Wilson networks in AdS and global conformal blocks

We develop the relation between gravitational Wilson line networks, defined as a particular product of Wilson line operators averaged over the cap states, and conformal correlators in the context of the AdS$_2$/CFT$_1$ correspondence. The $n$-point $sl(2, \mathbb{R})$ comb channel global conformal block in CFT$_1$ is explicitly calculated by means of the extrapolate dictionary relation from the gravitational Wilson line network with $n$ boundary endpoints stretched in AdS$_2$. Remarkably, the Wilson line calculation directly yields the conformal block in a particularly simple form which up to the leg factor is given by the comb function of cross-ratios. It is also found that the comb channel structure constants are expressed in terms of factorials and triangle functions of conformal weights whose form determines fusion rules for a given 3-valent vertex. We obtain analytic expressions for the Wilson line matrix elements in AdS$_2$ which are building blocks of the Wilson line networks. We analyze general cap states and specify those which lead to asymptotic values of the Wilson line networks interpreted as boundary correlators of CFT$_1$ primary operators. The cases of (in)finite-dimensional $sl(2, \mathbb{R})$ modules carried by Wilson lines are treated on equal footing that boils down to consideration of singular submodules and their contributions to the Wilson line matrix elements.

AdS vertex functions generally fixes the cap states to be the Ishibashi states. However, depending of particular values of weights which correspond to either irreducible or reducible and to either finite-or infinite-dimensional modules the cap states can be different. We list all admissible caps states and study their properties, all within a single framework. In particular, that allowed us to clarify the relation between the Wilson line construction in the finite-dimensional case elaborated previously in [10,[22][23][24] and in the infinite-dimensional case [8,15]. In order to relate AdS vertex functions in the bulk with CFT correlation functions and, in particular, with conformal blocks on the boundary we use the extrapolate dictionary [57][58][59][60]. As such the extrapolate relation defines a CFT correlation function as the near-boundary asymptotics of a given AdS vertex function. The respective conformal block arises after stripping off the structure constants which are calculated to be particular functions of weights. Instead of calculating the conformal blocks directly as the asymptotic AdS vertex functions we show that near the boundary the AdS vertex functions satisfy a recursion relation which expresses an n-point function through (n − 1)-point functions. This recursion relation can be explicitly solved in terms of the 4-point functions. In this way, we reproduce the recursion relation for conformal blocks in the comb channel originally observed in purely CFT terms within the shadow formalism [56]. In short, our asymptotic recursion relation is a mere reflection of that the Wilson network operator in the comb channel is the matrix product and adding two more legs to the (n − 1)-point comb diagram is realized by multiplying the respective Wilson matrix element by some typical matrix that extends it to the n-point comb diagram.
The paper is organized as follows. Section 2 reviews the gravitational Wilson line network construction in AdS 2 spacetime and sets our notation and conventions for sl(2, R) modules described in the ladder basis. Here, an AdS vertex function is introduced as the Wilson network matrix element with n endpoints. Following the extrapolate dictionary, in Section 3 we impose the sl(2, R) invariance condition on the AdS vertex functions that boils down to the cap state condition which we solve for various types of sl(2, R) modules. Here, we formulate the Wilson network/conformal block correspondence. Also, we introduce a weaker cap state condition which correctly captures only the near-boundary behaviour of AdS vertex functions. By this we mean that the strong condition is that the AdS vertex functions satisfy the sl(2, R) Ward identities in the bulk and, as a consequence, the asymptotic AdS vertex functions satisfy the conformal Ward identities on the boundary, while the weak condition requires the Ward identities only asymptotically. In Section 4, aiming to find conformal blocks we calculate Wilson line elements in a closed form and analyze their asymptotic behaviour for all admissible cap states. We show that the Wilson matrix elements are generally given by the hypergeometric functions 2 F 1 of complex arguments. Here, in particular, in order to demonstrate various peculiarities of our construction we consider an example of the asymptotic 2-point AdS vertex functions and respective 2-point CFT correlators. In Section 5 we derive the recursion relation satisfied by asymptotic AdS vertex functions and explicitly solve it with the base of recursion being the 4-point AdS vertex function. In Section 6 we summa-rize our results and discuss possible further developments. Appendix A collects a few explicit formulas for 3j symbols, as well as contains various special functions and their properties. Appendix B collects various detailed calculations. In Appendix C the SL(2, R) conformal transformation of the asymptotic AdS vertex functions are derived. Appendix D considers the lower-point AdS vertex functions and CFT correlation functions.

AdS 2 gravitational Wilson lines
We consider the zero-curvature condition dA + A ∧ A = 0 for sl(2, R) gauge connections A = A(ρ, z) on the two-dimensional manifold M 2 with local coordinates x µ = (ρ, z), where ρ, z ∈ R. The zero-curvature condition can be realized dynamically as the equation of motion following from the BF action where T r stands for the Killing invariant form, B is a scalar field, and F = dA + A ∧ A. The action describes a two-dimensional dilaton gravity with a non-zero cosmological constant [61][62][63] which is the Jackiw-Teitelboim model [64,65]. the solution of the zero-curvature condition can be cast into the form [66] A = e −ρJ 0 J 1 dze ρJ 0 + J 0 dρ . (2.3) The associated metric of the AdS 2 spacetime is given by where the conformal boundary lies at ρ = ∞. 3 The Wilson line is defined as where L is a path from x 1 to x 2 ; P is the path ordering operator; the index j means that A takes values in an (in)finite-dimensional module R j of sl(2, R) of weight j. The main properties of W j [L]: ] -a path transitivity; 3) W j [L] associated with a flat connection depends only on the endpoints of L. (2.6) 3 AdS2 spacetime has two conformal boundaries but the Poincare coordinates used here cover only one of them.
In order to calculate the Wilson line associated to the connection (2.3) one can gauge transform A with respect to the gauge element g(x) = e ρJ 0 and then use the gauge transformation property (2.6). One finds the following form of the Wilson operator which proves to be convenient in further calculations. Note that the whole consideration can be straightforwardly extended to the AdS 3 /CFT 2 case by introducing (anti)holomorphic coordinates z,z. The gauge algebra is then sl(2, R) ⊕ sl(2, R) and the Wilson line operators (2.5) are to be supplemented by anti-chiral Wilson operators which are associated with modules Rj of the anti-chiral algebra sl(2, R). The (anti)chiral factorization underlines the conformal block factorization of the boundary CFT 2 and (anti)holomorphic dimensions (h,h) are expressed in terms of (anti)chiral weights (j,j).

Group-theoretic conventions
The Wilson lines (2.7) can be combined to form a network with 3-valent vertices, see Fig. 1. The respective sl(2, R) modules meeting in vertices are related by 3-valent intertwiners, which are invariant tensors from Inv(R * j 1 ⊗ R j 2 ⊗ R j 3 ). The intertwiners have the obvious invariance property I j 1 j 2 j 3 U j 2 U j 3 = U j 1 I j 1 j 2 j 3 , (2.9) where U j are SL(2, R) operators of the corresponding representations. We will be interested in two types of sl(2, R) modules R j carried by Wilson lines, finiteand infinite-dimensional. Below we describe them in the ladder basis. (2.17)

AdS vertex functions
The matrix element of the Wilson line network can be directly read off from the comb graph on Fig. 1 (a) by moving from left to right: where the first line of operators describes inner edges connecting vertices with one external edge and the second line stands for the external edges except for the first one. Note that the 0 z ρ x 1 x 2 x 3 x 4 x 5 x n−1 x n y 1 y 2 Figure 1. The sets of endpoints and vertices will be denoted, respectively, as The endpoints x in the Wilson network operator (2.18) are so far arbitrary. However, from the CFT perspective they are to be taken on the boundary. Since the conformal boundary lies at ρ = ∞ it is convenient to place all the endpoints on a hypersurface of constant radial coordinate ρ, i.e. x = (ρ, z), where, eventually, ρ → ∞ (see Fig. 1 (b)), and the boundary coordinate set is z = (z 1 , ..., z n ) . (2.20) Let us now associate to each endpoint a particular cap state |a i ∈ R j i . Then, denoting the Wilson network operator (2.18) as W j 1 ...jñ j 1 ...j n−3 (x, y) one can introduce its matrix element as which we call an AdS vertex function. Using the intertwiner invariance property (2.9) one can directly show that the AdS vertex function is independent of positions of the vertices y i ∈ y that can be equivalently expressed by a convenient choice y = 0 [8,9,22]. Since the radial components of vertex points are zero, ρ i = 0, then it follows that any such point lies deep inside the bulk and not on or near the boundary (ρ ∞), see Fig. 2. Using that y = 0 we change notation for the AdS vertex function as x 5 x n 2 x n 1 x n y 1 y 2 y 3 y 4 y n 4 y n 3 1 Figure 2. The Wilson line network can be equivalently represented as a graph with a single n-valent vertex (red dot) defined by the n-valent intertwiner.
where z are the boundary points (2.20), and ρ labels the line which will be finally pulled at (conformal) infinity, ρ → ∞. By using the identity resolutions 1 = m |j, m j, m| the AdS vertex function (2.21) can be represented as a matrix product · · · j n−3 , p n−3 |Ij n−3 j n−1 jn |j n−1 , m n−1 ⊗ |j n , m n ã 1 |j 1 , m 1 j 2 , m 2 |ã 2 · · · j n , m n |ã n , (2.23) where we introduced summations over indices m = (m 1 , ..., m n ) and p = (p 1 , ..., p n−3 ) with m i ∈ J i and p k ∈J k in the finite-dimensional case (2.15) or m i ∈ J − i and p k ∈J − k in the infinite-dimensional case (2.16), as well as x-dependent cap states (2.24) Thus, the AdS vertex function is a contraction of the intertwiner matrix elements corresponding to the amputated comb diagram and particular matrix elements of the Wilson operators representing external legs. In fact, the amputated diagram is described by the n-valent intertwiner I j 1 ...jn|j 1 ...j n−3 for external/internal modules R j and Rj joined into the comb diagram so that the AdS vertex function takes the equivalent form [22]: where the n-valent intertwiner is given by see Fig. 2. The generalized intertwiner invariance property directly follows from (2.9): which can be written in the infinitesimal form as where m = 0, ±1 and the superscript in (J m ) j indicates that J m is taken in R j .

Spacetime invariance and cap states
The AdS/CFT correspondence is usually understood as the equality of AdS and CFT partition functions so that the correlation functions arise by differentiating the partition functions. On the other hand, a different way to state the correspondence is to extrapolate AdS correlation functions to the conformal boundary [57][58][59][60]. E.g. for AdS 2 scalar quantum fieldsΦ i (ρ i , z i ) with masses m i the extrapolate dictionary gives 4 where conformal dimensions ∆ i of CFT 1 primary operatorsÔ i (z i ) are related to masses as . Note that all AdS 2 fields are placed on the hypersurface ρ = const which eventually tends to the conformal boundary.
In our context the AdS vertex functions are assumed to reproduce CFT correlation functions in the way (see the relation (3.23) below) which is essentially the same as the extrapolate dictionary relation (3.1). However, the AdS vertex functions are not literally AdS scalar correlation functions so in order to draw a parallel between the Wilson network/conformal block correspondence and the extrapolate dictionary the AdS vertex functions must be subject to particular spacetime symmetry criteria that mimic those satisfied by AdS scalar correlation functions.

Symmetry condition
We require the AdS vertex functions to be invariant with respect to AdS 2 spacetime isometry transformations: Note that the AdS vertex functions are chosen on the ρ = const hypersurface [see the discussion below (2.19)] that accords with the extrapolate dictionary relation (3.1). The infinitesimal form of the symmetry condition is given by three Ward identities where J m = ξ µ m ∂ µ are the Lie derivatives along the Killing vector fields ξ m (x) of the AdS 2 spacetime with the metric (2.4), the superscript i indicates that the derivative is taken with respect to the i-th coordinate. The Ward identities (3.3) uniquely fix the form of the cap states |a used to build the AdS vertex functions. Below we show that the AdS 2 isometry condition boils down to the following condition imposed on the ket cap states: The conjugated cap state a| satisfies the same equation since (J ± ) † ↔ J ∓ . In fact, this condition defines the (twisted) Ishibashi state [67]. 5 Below we list various solutions to the cap state condition depending on which particular module R j was chosen.
• In the case j / ∈ N 0 /2 there is a unique (up to a normalization) vector |a ≡ |j ∈ D − j [68]: • The case j ∈ N 0 is to be considered separately because D − j contains a singular vector |j, −j − 1 ∈ D − −j−1 ⊂ D − j that additionally generates a new solution to the cap state equation (3.5). In other words, the kernel of J 1 + J −1 becomes two-dimensional. The cap state is |j = α|j 1 + β|j 2 , where α, β ∈ R and the two basis cap states read The first basis cap state has finitely many terms since (J 1 ) 2n acts trivially on the HW vector |j, j at n > j. Note that the second basis cap state |j 2 ∈ D − −j−1 ⊂ D − j and, therefore, it can be obtained from (3.6) by j → −j − 1. Then, the first basis cap state • In the case j ∈ N 0 + 1 2 the only solution is given by (3.8) (see Appendix B.1).
• The case of finite-dimensional modules D j with j ∈ N 0 directly follows from the previous analysis. The calculation here is almost the same as that for D − j with j ∈ N 0 and the resulting cap state is given by |j = |j 1 . In the case j ∈ N 0 + 1 2 the cap state condition has no solutions.
An equivalent way to specify the cap states is to use the approach of Nakayama and Ooguri [68,69] which invokes the symmetry argument to localize CFT operators in the dual AdS spacetime thereby guaranteeing the extrapolate dictionary relation (3.1) (see also earlier works [70,71]). To provide a link between the Nakayama-Ooguri construction and the present Wilson line construction one introduces an AdS 2 statê Φ(0, 0) |0 = |j , where |0 is a vacuum state . (3.10) According to the Nakayama-Ooguri construction this state satisfies the same condition (3.5) which now follows from adjusting AdS and CFT isometries (by the AdS/CFT correspondence one assumes that spaces of states of AdS and CFT theories are isomorphic). Then, shifting this state to any point in AdS 2 one finds one-particle stateΦ(ρ, z) |0 in the space of states of the scalar theory. This is the AdS 2 wave function satisfying the Klein-Gordon equation.
Recall that the one-particle states span an infinite-dimensional space isomorphic to D − j , where the weight j defines the mass m 2 = j(j + 1). From this perspective, the Wilson state (2.24) can be viewed as a wave function realizing one-particle states in the AdS 2 massive scalar theory. 6 Indeed, by construction it belongs to D − j and satisfies the AdS 2 Klein-Gordon equation with the same mass term m 2 = j(j + 1) [15,17]. Thus, going to the multi-particle states one concludes that the AdS vertex functions of the Wilson line networks realize scalar field correlators in AdS 2 that, in particular, justifies the symmetry condition (3.3).
The above discussion results in the following remarkable property satisfied by the AdS vertex function: ., x n ) = 0 , i = 1, ..., n , (3.12) where i is the AdS 2 d'Alembertian evaluated in x i -coordinates and m 2 i = j i (j i + 1). We assume that the boundary points are essentially distinct, x i = x j , so that there are no terms ∼ δ(x i − x j ) on the right-hand side. In this respect, the AdS vertex functions are similar to Wightman functions for free scalar fields but V jj (x) = 0 for odd n.

Bulk cap state condition
Let us now derive the cap state condition (3.5) from the symmetry condition (3.3). To this end, one recalls that the Wilson line operators are covariantly constant with respect to the endpoints, i.e. (compact, non-gravitational) Chern-Simons theory and their connection to conformal blocks [1][2][3]. where Then, the left-hand side of (3.3) takes the form: 16) where the AdS vertex function is taken in the form (2.25). Here, a superscript in (J l ) j indicates that J l are taken in R j . For m = 0 or m = −1 the coefficients ε m l (x) are xindependent so that one can factor them out and use the generalized intertwiner invariance property (2.28) to show that the right-hand side of (3.16) equals zero for any cap states. In the case m = 1 the coefficients ε m l (x) are x-dependent so the previous argument does not apply that means possible constraints to be imposed on the cap states. In fact, since variables x i are independent, the right-hand side of (3.16) equals zero iff the following relation holds whereε l are constants so that the right-hand side is the product of some constant sl(2, R) element and the Wilson line operator. In the sequel, the superscript in (J l ) j is omitted as we will be considering only R j . Then, using the relation (3.14) for m = 1 one finds that the left-hand side of the above condition is given by Combining the last two relations one finds that the coefficientsε and the cap state |a must satisfy the constraint l=0,±1ε To solve this equation one commutes J i and W j [x, 0] to obtain: Then, acting with the inverse Wilson operator W j [0, x] and equating the coefficients in front of the same powers of z and e ρ one finds that the relation (3.17) is valid provided that To summarize, we showed that the Ward identities for the AdS vertex functions are guaranteed by the following relation which becomes an identity provided the cap state satisfies (3.5). Note that this relation is valid for both (in)finite-dimensional modules.

sl(2, R) conformal blocks
Invoking the extrapolate dictionary relation (3.1) one can now explicitly relate n-point global CFT 1 conformal blocks in the comb channel which we denote by F hh (z) (see Fig. 3) to the n-point AdS vertex functions V jj (z, ρ). To this end, one chooses the cap states |a i as in Section 3.1 and relates conformal dimensions with weights as h i = −j i andh k = −j k . 7 Then, the exact relation reads The normalization coefficients C jj ≡ C j 1 ...jnj 1 ...j n−3 are given by where ∆(a, b, c) = (a+b+c+1)!(a+b−c)!(a+c−b)!(b+c−a)! is the modified triangle coefficient and we assume that for arbitrary real arguments the factorials are represented as x! = Γ(x+1).
Note that C j 1 j 2 = C j 1 j 2 0 which means that C j 1 j 2 j 3 defines all the normalization coefficients in (3.24). The pole in C j 1 j 2 at j 1 = − 1 2 is an artefact of our choice of the normalization of the intertwiner (2.17). 8 In order to have non-vanishing and real C j 1 j 2 j 3 one restricts the weights via triangle inequalities: 1) j 1 , j 2 ∈ N 0 /2, j 3 ∈ Z : 2) in other cases : In fact, these conditions derived here by analysing the singular points of factorials in (3.24) completely reproduce the triangle identities coming from the Clebsch-Gordan series for R j 1 ⊗ R j 2 projected on R j 3 depending on the choice of particular weights. Equivalently, they are encoded in the 3j symbol, see (A.1). The conformal blocks are normalized to contain the leg factors which are particular scale prefactors depending on z that ensure correct conformal transformation properties. With the leg factor stripped off the conformal block is a function of cross-ratios only (the so-called bare block). On the other hand, the coefficients C j 1 j 2 j 3 are in fact the structure constants of 3-point correlation functions and, therefore, C jj is a product of the structure constants which arise when decomposing an n-point correlation function into conformal blocks in the comb channel. Thus, we conclude that the AdS Wilson line networks yield CFT correlation functions with specified structure constants expressed in terms of conformal weights (3.24).

Boundary cap state conditions
Suppose now that we are not interested in interpreting the AdS vertex function as an independent and meaningful object in the bulk but only as an auxiliary function which can be used to calculate conformal blocks in the large-ρ limit. It is clear then that one can choose other cap states which guarantee the Ward identities only near the boundary. Obviously, they would satisfy some conditions which are weaker than the cap state condition in the bulk (3.5). Indeed, requiring conformal symmetry of the AdS vertex function only at large-ρ and using the arguments similar to those discussed in the previous sections one finds a system of asymptotic equations where the left-hand sides are given by x-dependent vectors in R j which are sub-leading at large-ρ. Here, |ã : which is more convenient for finding solutions. In the sequel, we call such states as quasi-Ishibashi cap states. We are not going to describe a general solution to the system of asymptotic equations (3.28) and (3.29), instead, below we list some simple partial solutions.
• For infinite-dimensional modules D − j the asymptotic equations are solved by two (generally non-conjugated) bra and ket vectors whereR,Ř ∈ U (sl(2, R)) are some constant elements of the universal enveloping algebra of sl(2, R). Their form is fixed by the large-ρ behaviour read off from (3.28)-(3.29).
Obviously, both the Ishibashi states (3.6) for j / ∈ N 0 /2 and (3.7)-(3.8) for j ∈ N 0 are partial solutions. Yet another partial solution is given by the state where j ∈ R and the basis vectors |j, j − n ∈ D − j . In the last equality, this cap state is represented by e −J 1 rotation of the HW vector for which reason we name it as a rotated HW state. One may notice that it satisfies the condition (J 1 − J −1 + 2J 0 ) |j = 0.
• For finite-dimensional modules D j the operator (J 0 + j) in (3.29) has a kernel described by LW vectors (recall that (D j ) * ≈ D j ) so that in this case the general solution can be represented as with some new constantL,Ľ ∈ U (sl(2, R)). One partial solution is given by (3.9). It immediately follows that the LW/HW vectors proposed in [9] as the cap states solve the above asymptotic equations (i.e.L,Ľ = 0), We emphasize that neither (3.31) nor (3.33) satisfy the cap state condition (3.5) in the bulk. That, in its turn, means that the respective AdS vertex function is not sl(2, R) invariant (3.2). However, if we aim at just reproducing conformal blocks as the large-ρ asymptotics then such a choice is admissible. This is not surprising since controlling sl(2, R) invariance only near the boundary one can find more possible cap states. However, if one wants to reconstruct bulk correlators then the AdS vertex functions and the cap states are obliged to satisfy the formulated symmetry conditions. This is exactly in the spirit of the HKLL bulk reconstruction [72,73].
For convenience, we collect the (quasi)-Ishibashi states in Table 1. We see that for particular values of weights there are no solutions to the cap state condition in the bulk (3.5), though the cap state conditions near the boundary (3.28)-(3.29) always have non-trivial solutions. Presumably, the absence of solutions means that the Ward identities imposed on the AdS vertex functions must be modified by adding spin parts. It means that the cap state condition is also modified by spin terms, for a discussion see [15,68]. We hope to consider this issue elsewhere.

Wilson line matrix elements and their asymptotics
In order to calculate CFT correlation functions according to the extrapolate dictionary relation (3.23) one has to analyze the large-ρ behaviour of the AdS vertex functions. Since a given AdS vertex function is built from the Wilson matrix elements then its large-ρ dependence is completely defined by their asymptotic expansions. Knowing the large-ρ asymptotics of the Wilson matrix elements allows assembling conformal correlation functions directly as their products with the respective n-valent intertwiner which guaranties an invariant contraction of indices in different representations.
The key observation here is that the Wilson matrix elements j, m|W j [0, x]| j and j|W j [0, x] |j, m (we call them right and left according to whether the cap state is ket or bra) have different behaviour when approaching the boundary from the bulk. More precisely, the right elements are well defined near the boundary, while the left elements diverge. The Wilson matrix elements are built as infinite sums which is a natural consequence of that in the ladder basis the cap state is an infinite linear combination of basis vectors, hence, this infinite summation is inherited in the Wilson matrix elements. To cure the divergent behaviour of the left Wilson elements we find them in a closed form that allows one to analytically continue in ρ and finally expand near the boundary.
Following the classification of the cap states in Table 1 below we list all exact Wilson matrix elements ã|j, m = j|W j [0, x] |j, m and j, m|ã = j, m|W j [0, x]| j .
Proposition 1 Denote q = −ze ρ . The left and right Wilson matrix elements are given by: (4.5) • for the LW/HW states in D j , Now that we have exact expressions for the Wilson matrix elements we can analyze their leading large-ρ asymptotics. One finds out that regardless of what particular cap state (Ishibashi or quasi-Ishibashi) of one or another module (finite-or infinite-dimensional) is used the asymptotic Wilson matrix elements are the same (modulo signs): where the overall coefficient is given by (4.5) and the parameter δ in the left Wilson matrix element distinguishes the two cases: (1) δ = 2j when j| is the Ishibashi states (3.6) or (3.7)-(3.8); (2) δ = j when j| is the rotated LW (3.31) or δ = j when j| is the LW vector. The symbol ≈ denotes keeping leading large-ρ contributions only. Also, in the case of LW/HW cap states the subleading terms are absent.
Let us now discuss for which j and m the overall coefficient may have singular points. To this end, one notes that A jm can be represented in terms of the coefficient function M (j, m) of the ladder basis (2.13) as As one can see, for j ∈ N 0 /2 and m < −j one of the factors in the product becomes zero. Since the zeros correspond to singular vectors (see our comment below (2.13)) we conclude that the domain of m is effectively restricted to m ∈ [[−j, j]] corresponding to a finite-dimensional D j . On the other hand, A jm has no poles. It means that we can safely write the coefficient (4.5) without specifying the domain of m since the coefficient itself efficiently fixes this domain depending on the weight j of a given module R j . In other words, the reason is that A jm is expressed in terms of the gamma-functions z! = Γ(z + 1) which effectively control a transition between finite-and infinite-dimensional cases: in one direction -through the analytic continuation of the factorials to the gamma-functions; in the opposite direction -through the simple poles of the gamma-function. Thus, the formulae (4.7), (4.8), and (4.9) explicitly demonstrate the following: The right and left Wilson matrix elements in D j can be analytically continued to the leading asymptotics of the right and left Wilson matrix elements in D − j (modulo overall signs) by extending weights j from integers to reals. In the next sections we calculate: (1) the Wilson matrix elements in D − j with j / ∈ N 0 /2 (Sections 4.1-4.2); (2) the Wilson matrix elements in D − j with j ∈ N 0 (Sections 4.3); (3) the Wilson matrix elements in D j with j ∈ N 0 (the final comment in Section 4.3). The boundary consideration of the Wilson matrix elements for the quasi-Ishibashi states of Section 3.4 is basically the same as that for the true Ishibashi states and, therefore, this analysis is completely relegated to Appendix B.6.

Right Wilson matrix element
In Appendix B.2 we obtained the right Wilson matrix element in the form (B.12): This function is a polynomial in q = −ze ρ meaning that its radius of convergence in q is infinite. Evaluating the sum in (4.10) we find that the right Wilson matrix element can be represented in a closed form (B.17): (4.11) Expanding this function near |q| = ∞ and substituting back q = −ze ρ one finds cf. (4.8).

Left Wilson matrix element
The left Wilson matrix element for the cap state (3.6) in the form of the power series is calculated in Appendix B.2, where we found the expression (B.13): (4.13) Contrary to the right Wilson matrix element this one is given by an infinite power series in variable −ze ρ (modulo prefactors). Thus, one is obliged to analyze the issue of convergence of the following power series q = −ze ρ : n a n q n , where a n = (−i) (4.14) Using e.g. the d'Alembert's ratio test one can show that the radius of convergence equals one, i.e. |q| < 1, see Appendix B.4. In terms of ρ-coordinate one has ρ < − log |z|, which means that for arbitrary z the radius of convergence in ρ goes to zero. Nonetheless, below we show that the function can be analytically continued past |q| = 1 thereby making the large-ρ expansion possible. As shown in Appendix B.5 the power series (4.13) can be summed up to yield the Wilson matrix element in a closed form (B.25): By construction, this function is still defined inside the disk |q| = 1 (note that the factor e −ρm is passive here). In order to analytically continue beyond |q| = 1 one notes that the second parameter of the hypergeometric function m−j is a negative integer. It means that the hypergeometric function is a polynomial in the Cayley transformed variable q−i q+i with degrees running from 0 to j − m. Any polynomial is a holomorphic function and being originally defined in a domain it can be analytically continued to the whole complex plane.
Next, note that the whole expression (4.15) is proportional to j−m k=0 α k (q+i) j−k (q−i) m+k with some α k . At j ∈ Z (then m is also integer as follows from (2.12)) the powers of q + i and q − i are integer, therefore, (4.15) can be analytically continued without any branch cuts. At j / ∈ Z, (4.15) has three branching points (coming from the prefactor (q + i) j (q − i) m ): q = i, q = −i, and q = ∞. Choosing branch cuts along the imaginary axis Im q as (+i, +i∞) and (−i, −i∞) one analytically continues the left Wilson matrix element onto the whole real axis Re q.
Upon analytic continuation one sets q to be real again (along with z and ρ). Finally, one can expand the analytically continued expression (4.15) near |q| = ∞, substitute q = −ze ρ and obtain cf. (4.7).

Positive integer weights
In the case of D − j with j ∈ N 0 there are two independent caps states |j 1 and |j 2 , see (3.7), (3.8). As we discussed below (3.8) the cap state |j 2 solves the cap state condition for D − −j−1 and can be obtained from the cap state |j ∈ D − j by shifting j → −j − 1. It follows that the Wilson matrix elements with |j 2 can be directly obtained from (4.15) and (B.17) by using the same shift. Expanding these matrix elements near ρ = ∞ one obtains:  where α n , β n are some coefficients. In other words, they belong to a subspace orthogonal to the cap state |j 2 . Hence, the Wilson matrix elements (4.17) equal zero at m ≥ −j.
Consider now the cap state |j 1 (3.7). The process of calculating the right Wilson matrix element j, m| W j [x, 0]|j 1 is exactly the same as in the case j / ∈ N 0 /2 (4.10). To this end, we change the summation domain in (4.10) from n ∈ [[0, ∞]] to n ∈ [[0, 2j]] according to the definition of the |j 1 (3.7) and write where we used that j, m| (J 1 ) p+n |j, j is non-zero only for p+n = j −m. From the restriction One can see that the resulting expression coincides with (4.10) so the right Wilson matrix element at j ∈ N is given by (4.11): Similarly one computes the corresponding left Wilson matrix element. Taking (4.13) and introducing a new summation domain one obtains: Since the summation domain is finite, then the radius of convergence for q is infinite and (4.21) is defined for any real ρ and z. The final answer is the same as in (4.15), (4.16): We conclude that for j ∈ N 0 the matrix elements (4.17) with |j 2 are suppressed by the matrix elements (4.20) and (4.22) with |j 1 . In this way, we see that the degeneracy of the cap states |j = α|j 1 + β|j 2 (3.7) is lifted and it is the cap state |j 1 which defines the boundary behaviour of the Wilson matrix elements which form conforms with the general formula (4.7), (4.8). Simultaneously, in the case of j ∈ N 0 + 1 2 the Wilson matrix elements are defined with respect to |j 2 only and, therefore, they decay near the boundary much faster than assumed by the extrapolate dictionary relation (3.23). Therefore, in this case the boundary CFT correlation function vanishes.
The form of the Wilson matrix elements in D j directly follows from the above analysis. In this case, the cap state is given by |j 1 ∈ D j (3.9) and the boundary behaviour of the Wilson matrix elements in D j coincides with that of the Wilson matrix elements in D − j with j ∈ N 0 up to a sign, see (4.7), (4.8).

Conformal invariance
We now show that near the boundary the Ward identities for AdS vertex functions go to the Ward identities for CFT correlation functions. Using the right matrix element j, m|ã asymptotics (4.8) one directly finds how the AdS 2 Killing generators are restricted on the boundary: J n j, m|ã = L n j, m|ã + O(e ρ(j−1) ) , ∀ j, m| , (4.23) where L n = z n+1 ∂ z − j(n + 1)z n , n = 0, ±1 .
where the superscript i indicates that the differential operator is taken with respect to the i-th coordinate. Taking the limit ρ → ∞ and using the identification with CFT 1 correlation functions (3.23) one finds out that the above relation goes into the sl(2, R) conformal Ward identities.
In Appendix C we also show that n-point AdS vertex functions (2.23) are conformally invariant against finite transformations, i.e. they change under SL(2, R) : z → w(z) as Using the extrapolate dictionary relation (3.23) one obtains the conformal transformation law of n-point CFT correlation functions of primary operators.
where the Wilson matrix elements are given by the asymptotics (4.7) and (4.8), and the 2valent intertwiner ∈ Inv (D − j 1 ) * ⊗ D − j 2 can be directly read off from (2.17) by substituting j 3 = 0: Summing over m 2 along with changing k = j 1 − m 1 yields Using the generalized Newton theorem (A.4) one finally obtains where in the last equality we used j 1 = j 2 = −h and introduced the normalization constant C = [1 − 2h] − 1 2 that through the identification (3.23) gave us the 2-point CFT correlation function.
So far we have been controlling only the large-ρ behaviour of the Wilson matrix elements. However, these also depend on z-variables and combining them together brings to light the 9 Originally, this property was established for 4-point functions [9] in the case of finite-dimensional modules (for quasi-Ishibashi states, in the present terminology). In Appendix C we show this property for n-point functions in the case of infinite-dimensional modules. See [14,16,22] for more discussion in the present context and [15] for the symmetry analysis in d dimensions.
issue of convergence of the AdS vertex functions in z-space. Indeed, for the series (4.30) to be convergent the boundary points must be ordered as z 1 > z 2 . The same ordering prescription z 1 > z 2 > . . . > z n (4.32) persists for higher-point AdS vertex functions that yields a particular OPE ordering for CFT correlation functions giving rise to the conformal block decomposition in the comb channel.
Remarkably, the convergence and ordering issues are absent for the AdS vertex functions in the finite-dimensional modules since all functions in this case are polynomials both in z and ρ coordinates. Finally, note that since (z − 1)! ≡ Γ(z) has simple poles in z = 0, −1, −2, . . . one finds out that for 2j 1 ∈ N 0 all terms in (4.30) with k > 2j 1 vanish. Of course, this is just a direct consequence of Proposition 2 since the AdS vertex function is built as the product of left and right Wilson elements. Thus, the infinite series for (half-)integer weights keeps only finitely many terms. This yields the 2-point function for finite-dimensional modules which can be equivalently calculated using the HW/LW vectors as the cap states. The only difference is given by the overall sign that follows from Proposition 3. This observation extends to the n-point case that allows us to simplify all calculations by using the finite-dimensional irreps and then analytically continue in weights. We use this trick in the next section.

Higher-point functions and recursion relations
As discussed in the previous section, by virtue of Propositions 2 and 3 we can restrict our consideration to finite-dimensional modules. The n-point conformal blocks will be calculated by solving the recursion relation satisfied by the asymptotic AdS vertex functions (though, for finite-dimensional modules these functions are exact in ρ). The base of recursion is given by the 4-point conformal block which along with 3-point and 5-point blocks is calculated in Appendix D.

Recursive construction of AdS vertex functions
Here and later, we use the following notation for coordinate sets: According to the matrix expression of the AdS vertex functions (2.23), the (n − 1)-point function for a set of finite-dimensional modules D j and Dj can be represented as x n 3 x n 2 x n 1 x n y 1 y 2 y 3 y n 5 y n 4 y n 3 1 Figure 4. From n − 1 points (left) to n points (right). The blue graph stands for the auxiliary matrix element C pn−4 ∈ Dj n−4 (5.4). Dotted green lines denote those parts of the diagrams which are affected by adding one more endpoint. and the n-point function as  3) into two parts by singling out the terms contained in the first and second lines. Indeed, let us define the auxiliary matrix element which is the same for both expressions. It has one external index in Dj n−4 module. Of course, this matrix element has less dummy summation indices since it describes just a sub-diagram (the same for both diagrams) connected to other edges by additional summations.
The technical reason for such a factorization is that the last two legs of the (n − 1)-point Wilson line network carrying D j n−2 and D j n−1 are connected to the rest of the diagram only by intertwiner Ij n−4 j n−2 j n−1 which arises in the second line of (5.2). Extending (n − 1)-point diagram to n-point diagram by adding two more legs (one intermediate Dj n−3 and one external D jn ) results in replacing Ij n−4 j n−2 j n−1 by Ij n−4 j n−2jn−3 which correspond to replacing external D j n−1 by internal Dj n−3 and adding one more intertwiner Ij n−3 j n−1 jn arsing in the third line of (5.3). This procedure is depicted on Fig. 4. In what follows we explicitly single out C p n−4 (ρ, z ) in both AdS vertex functions that allows us to find out that they are recursively related.

(5.5)
To simplify further calculations we make use of the following coordinate SL(2, R) transformation where we introduced the large coordinate parameter ω → ∞ to regularize the pole z = z n−2 in (5.6). We want to simplify (5.5) without calculating explicitly the auxiliary matrix element C p n−4 (z ). The most efficient way to do that is to write the AdS vertex function in wcoordinates, take the last two points w n−2 and w n−1 as ∞ and 0 because the Kronecker symbols arising in the corresponding Wilson matrix elements will drastically simplify the intertwiner and then make an inverse transformation to (5.6) to obtain the simplified AdS vertex function in z-coordinates. Namely, the last two Wilson matrix elements (4.6) in wcoordinates are given by j n−2 , m n−2 |ã n−2 = e ρj n−2 δ −j n−2 ,m n−2 ω 2j n−2 + O(ω 2j 3 −1 ) , j n−1 , m n−1 |ã n−1 = e ρj n−1 δ j n−1 ,m n−1 , where we used that the prefactor A jm trivializes: A jj = A j−j = 1. In the first matrix element we kept only the leading asymptotics in ω. Indeed, one can see that the AdS vertex function (2.23) is a polynomial in ω of powers running from 0 to 2j n−2 and the leading contribution at ω − → ∞ is provided by the maximal power (j n−2 ∈ N). The Kronecker delta in the second matrix element appeared because the right Wilson matrix element (4.6) is non-zero for z = 0 only when m = j. Then, using the Wilson matrix elements (5.8), resolving the Kronecker deltas by summing over m n−2 and m n−1 and using the 3j symbol (A.2) 10 one eventually finds the (n − 1)-point AdS vertex functions in w-coordinates: where the w-coordinate sets are w = (w 1 , ..., w n−4 , 1, ω, 0) and w = (w 1 , ..., w n−4 , 1) and the structure constant Cj n−4 j n−2 j n−1 is given by (3.24). For the sake of simplicity, here and later we suppress the subleading terms O(ω 2j−1 ). Now, making the inverse coordinate transformation z(w) = −z n−2 z n−3 n−1 w + z n−1 z n−3 n−2 −z n−3 n−1 w + z n−3 n−2 (5.10) and using the conformal transformation law (4.27) as well as taking the limit ω → ∞ we get , (5.11) where w (z ) = (w 1 (z ), ..., w n−4 (z ), 1, ω, 0). The power-law prefactors in the first equality are the Jacobians from (4.27).

Recursion relation
The previously obtained expressions for the AdS vertex functions demonstrate that a given n-point AdS vertex function (5.13) can be represented in terms of (n − 1)-point AdS vertex functions (5.11) with running last external weight. Indeed, expressing the auxiliary matrix element C form the relation (5.9) (which contains no summations) through the (n − 1)-point AdS vertex function and substituting it into (5.13) (which contains just one summation over k) along with the change of the summation index k we find the following relation The 3-point AdS vertex function Vj n−3 j n−1 jn on the right-hand side just encodes power-law prefactors in z. Here, we assume that the base of recursion is n = 5 which means that the first non-trivial recursion relations expresses the 5-point function on the left-hand side in terms of 4-point functions on the right-hand side. 11 Note that at n = 3 this recursion relation breaks up. However, it is still valid at n = 4, but needs replacingj n−4 with j 1 . This is why the more convenient base of recursion is n = 5. Here, we emphasize again that the recursion relation holds for asymptotic AdS vertex functions which coincide with their bulk expressions only for finite-dimensional modules.
(5.17) This form comes from using explicit expressions for the 3-point AdS vertex function and structure constants in (5.14). The coefficient in the recursion relation (5.16) is split in two parts: (1) the factor β n,kn which is independent of j n ; (2) the factor γ n,kn,jn which is dependent on j n . All the vertex functions and coefficients in (5.16) still depend on other weights and coordinates but here we indicate only those which are relevant for the recursion procedure.
At the next step of recursion, n → n + 1, the relation (5.16) reads Substituting the n-point AdS vertex function (5.16) into the right-hand side of (5.18) one obtains Considering n = 4 as the base of recursion one can directly see that the final recursive solution is given in terms of summing over 4-point AdS vertex functions with running last external weight, where the summation domains are  Recursive solution. In order to explicitly evaluate the products and sums in (5.20) it is convenient to use the obvious identity β n,kn n−1 i=5 β i,k i = β 5,k 5 n−1 i=5 β i+1,k i+1 and rewrite (5.20) as The right-hand side of this expression can be schematically represented as γ(βV)( βγ). In what follows we calculate each factor in this formula separately. Since the γ-factor here is explicitly given by (5.17) then we focus on the βV-factor and the βγ-factor. βV-factor. From Appendix D.2 we know that the 4-point AdS vertex function V j 1 j 2 j 3 (j 2 −k 5 )j 1 which is present in (5.22) is given in terms of the hypergeometric function, see (D.17). Using the hypergeometric series (A.5) one represents it as Substituting β 5,k 5 given by (5.17) we find the (βV)-factor: (5.26) βγ-factor. By means of (5.17) the βγ-product is explicitly calculated to be 27) where we used the obvious identity Up to the structure constants this expression is the n-point global conformal block in the comb channel [56].

Conclusion
In this work we have been studying the AdS 2 /CFT 1 correspondence as the Wilson network/conformal block correspondence. Having defined the AdS vertex functions as the Wilson network operators sandwiched between the cap states we have calculated the n-point global conformal blocks in the comb channel through the extrapolate dictionary relation. In particular, our results include: • Extrapolate dictionary relation for the AdS vertex functions and CFT correlation functions. To some extent, in this paper we have shifted the focus from calculating the conformal blocks on the boundary to the study of the bulk properties of the AdS vertex functions. In particular, we have shown that there is a family of AdS vertex functions which have the same boundary behaviour but only one of them is sl(2, R) invariant in the bulk.
• Cap states in (in)finite-dimensional modules in Table 1 and exact expressions for the Wilson matrix elements in Proposition 1. Also, we have formulated Propositions 2 and 3 about their large-ρ asymptotics that demonstrate a universal behaviour of the Wilson matrix elements in different sl(2, R) modules near the AdS boundary.
• Explicit calculation of n-point global conformal blocks via Wilson line networks. We have shown that the AdS vertex functions near the boundary satisfy a simple recursion relation that can be explicitly solved to give conformal blocks in the form previously known on the CFT side.
These constructs can be straightforwardly extended to the case of AdS 3 /CFT 2 in the spirit of Refs. [8,10,15,[22][23][24] just because of the (anti)chiral factorization underlying both the sl(2, R) × sl(2, R) Chern-Simons theory and the boundary CFT 2 . It would be interesting to extend the study of two-point AdS 3 vertex functions [17,21] to the multipoint case. Exact formulas for the Wilson matrix elements obtained in this paper can be useful in achieving this objective. Yet another interesting direction is to elaborate the Wilson network approach in AdS 3 spacetimes with conformal boundaries being genus-g Riemann surfaces (the present case is g = 0) that would allow one to formulate analogous near-boundary recursion relations and find closed-form formulas for genus-g global conformal blocks.
Acknowledgements. We are grateful to Semyon Mandrygin and Mikhail Pavlov for useful discussions.
A 3j symbol and special functions 3j symbol and intertwiner. The 3j symbol is defined as the matrix element of the 3-valent intertwiner (see e.g. [74]) At a = j a the 3j symbol is drastically simplified, The 3-valent intertwiner for finite-dimensional modules is given by [75] [ for j 1 , j 2 , j 3 ∈ N/2. In [75] it was shown that the 3-valent intertwiner for discrete series D ± j is the analytical continuation of (A.3) to any real weights j 1 , j 2 , j 3 . Throughout the paper we denote Γ(z + 1) = z! for any z.
Binomial expansion. The generalized Newton binomial is given by where a ∈ R and |x| < |y|.
Hypergeometric function. The hypergeometric series 2 F 1 (a, b; c|z) is given by where (p) k = Γ(p + k)/Γ(p) is the Pochhammer symbol and |z| < 1. The series can be analytically continued to the whole complex plane with the branch cut (1, ∞) by means of the Euler integral representation. The hypergeometric function at different (small/large) values of z can be related by the Pfaff transformation For z = −1 and c = a 1 − a 2 + 1, the hypergeometric function can be represented as Lemma. Using (A.5) and (A.6) one can prove the following relation: where j a , j b , j c ∈ N/2 and n ∈ Z, the summation domains are Appell function. The second Appell function is defined as which generalizes those for the hypergeometric and Appell functions given above. Another identity for the comb function is given by (A.16) This can be shown by using the following symmetry property of the comb function, 2) which is equivalent to the following system One can see that f 1 = f 3 = .

B.2 Another form of the Ishibashi state
In order to calculate the Wilson matrix elements we represent the Ishibashi state (3.6) in a more convenient form (the conjugated state j| is obtained by replacing J 1 → J −1 ). Recall that for notational convenience we replace all the gamma-functions by factorials according to Γ(x + 1) = x!. First, consider the following chain of identities The k-th term here reads These two types of terms with k = 0, ..., n cancel each other in (B.5) so that The prefactor in (B.5) is finite for j ∈ R, whence, the whole expression (B.5) equals zero, i.e.
One can notice that (B.9) is obtained from (B.4) by replacing 2n with 2n + 1. It means that we rewrite the coefficient in the Ishibashi state (3.6) in the form (B.4) and add the zero expressed through the hypergeometric series. Then, we use the identity (B.9) to obtain (B.10) Unifying the two sums above into a single sum one represents the Ishibashi state as 12 Note that despite the presence of i n the coefficients here are still real since all imaginary terms sum up to zero. Using this new form of the Ishibashi state the right Wilson matrix element can be written as The left Wilson matrix element for the conjugated Ishibashi state can now be rewritten as

B.3 Proving a closed-form of the right Wilson matrix element
The calculation of the right Wilson matrix element (B.12) proceeds by representing the hypergeometric coefficients as (A.5) and changing n = −k + j − m: where, for convenience, we introduced a new variable q = −ze ρ . Representing the sum over k as the hypergeometric series (A.5) and making the Pfaff transformation (A.6) yields (B.15) In its turn the hypergeometric function in the second line can be represented as (A.5) with a summation parameter k. Then, by changing k = −n + j − m − t one obtains (B.16) After using the generalized Newton binomial (A.4) and representing the sum over n as the hypergeometric function by means of (A.5) one finally finds: Note that the summation domains on each step of the calculation are finite. Thus, the variable q is not restricted by the condition of convergence so the right Wilson matrix elements are well-defined for any real q, in particular, near |q| = ∞.

B.4 Radius of convergence
Let us use the ratio test to find the radius of convergence of the power series (4.14): which by means of the identity (A.7) can be cast into the form Then, using the Stirling's approximation n! ∼ √ 2πn (n/e) n one finds (B.20) Thus, the radius of convergence does not depend on m and j so that any left Wilson matrix element converges for |q| < 1, where q = −ze ρ .

B.5 Proving a closed-form of the left Wilson matrix element
Below we make a few resummations that allows us to find a closed-form formula for the left Wilson matrix element (B.13). Rewriting the hypergeometric coefficient as (A.5) with a new summation parameter t and changing n = k + t one finds: Representing the sum over k as the hypergeometric series with |q| < 1 and making the Pfaff transformation (A.6) yields The argument of the hypergeometric coefficient here satisfies | iq iq−1 | < 1 for any real q. It follows that one can again represent it as (A.5) with a new parameter k: After changing t = s − k − m + j and using the generalized Newton binomial (A.4) (with |q| < 1) one sums over s to obtain: Finally, one sums over k in the second line to obtain the hypergeometric series. The sum is finite and, hence, converges for any q ∈ R (possible poles q = ±i are not on the real axis). Thus, we have a closed-form formula for the left Wilson matrix element: The resulting expression is real despite the presence of complex-valued arguments that can be explicitly seen by complex conjugation. In Section 4.2 we analytically continue this function past |q| = 1.

B.6 Wilson matrix elements for quasi-Ishibashi states
Rotated HW state. In this case, the calculation is much easier than that for the Ishibashi states. To this end, one represents the rotated HW cap state (3.31) as The conjugated state j| is obtained by replacing J 1 → J −1 . The right Wilson matrix element is given by where we changed the summation parameter n = t − m + j when going from the second line to the third line in which we used the notation q (4.14). The resulting sum has a finite radius of convergence, |q| < 1, whence, the matrix element is well-defined for small |q| only. In order to find the large-ρ asymptotics we analytically continue (B.29) as follows. For small |q|, using the generalized Newton binomial (A.4) one can explicitly sum over t to obtain the following closed-form expression: These relations are exact in ρ.
Finally, note that the finite-dimensional matrix elements (B.32), (B.33) can also be obtained from the rotated HW cap state analysis. Indeed, our consideration below (B.30) applies for j + m / ∈ N 0 . When j + m ∈ N 0 the summation domain of t in (B.29) becomes finite as the binomial coefficient (j+m)! t!(j+m−t)! equals zero for t > j + m. This means that there is no need to analytically continue function (B.30) and one can directly expand near ρ = ∞. The condition j + m ∈ N 0 along with the obvious relation j − m ∈ N 0 (2.12) leads to j ∈ N 0 /2, i.e. in this case we effectively deal with finite-dimensional modules. Note, however, that the singular submodule is not seen in this picture since the respective matrix elements are sub-leading. As we learned in Section 4.3 the singular cap state contributions are suppressed near the boundary. On the other hand, the quasi-Ishibashi states correctly capture only near-boundary effects.

B.7 Rearranging the n-point AdS vertex function
For the sake of simplicity, in the n-point case we make the same transformation (5.6) as in the (n − 1)-point case. Recalling that the inverse transformation (5.10) produces the Jacobians in the AdS vertex function (4.27) one observes that using the same coordinate map for the n-point and (n − 1)-point AdS vertex functions yields almost the same Jacobian factors.
The only difference is that the n-point AdS vertex function gets an additional ∂z ∂w jn w=wn because of the n-th point w(z n ). Since we aim to represent the n-point AdS vertex function in terms of (n − 1)-point ones by means of a recursion relation, then there is no need to consider coinciding factors in both expressions. Substituting (j n−4 − j n−1 − m n + j n−2 )!(j n−3 +j n−4 − j n−2 )!(j n−3 + j n − j n−1 )! (j n−3 −j n−4 + j n−2 )!(j n−4 + j n−2 −j n−3 )!(j n−4 + j n−2 +j n−3 + 1)!(−j n−3 + j n + j n−1 )!