Lattice walk area combinatorics, some remarkable trigonometric sums and Ap\'ery-like numbers

Explicit algebraic area enumeration formulae are derived for various lattice walks generalizing the canonical square lattice walk, and in particular for the triangular lattice chiral walk recently introduced by the authors. A key element in the enumeration is the derivation of some remarkable identities involving trigonometric sums --which are also important building blocks of non trivial quantum models such as the Hofstadter model-- and their explicit rewriting in terms of multiple binomial sums. An intriguing connection is also made with number theory and some classes of Ap\'ery-like numbers, the cousins of the Ap\'ery numbers which play a central role in irrationality considerations for {\zeta}(2) and {\zeta}(3).


Introduction
The enumeration of random walks of given algebraic area on a two-dimensional lattice is a hard and challenging problem.The algebraic area is defined as the oriented area spanned by the walk as it traces the lattice.A unit lattice cell enclosed in the counterclockwise (positive) way has an area +1, whereas when enclosed in the clockwise (negative) way it has an area −1.The total algebraic area is the area enclosed by the walk weighted by the winding number: if the walk winds around more than once, the area is counted with multiplicity.The combinatorics of such walks depend on the exact rule generating them and on the lattice geometry.The canonical example is closed random walks on a square lattice.This problem can be mapped to the famous Hofstadter model [2] of a particle hopping on a square lattice pierced by a constant magnetic field, with the value of the magnetic field playing a role analogous to the chemical potential for the area of the walk.Indeed, algebraic area enumerations are mapped on quantum mechanical models since in quantum mechanics a magnetic field couples to the area spanned by the particle.
An exact formula for the number of square lattice walks of given length and algebraic area was only recently obtained in the form of nested binomial sums [1].The analysis revealed some remarkable trigonometric sums to be key ingredients for the algebraic area enumeration.They are defined for p and q coprime positive integers as 1 q q k=1 b p/q l 1 (k)b p/q l 2 (k + 1) . . .b p/q l j (k + j − 1) where b p/q (k) is a trigonometric function called spectral function which depends on the rational number p/q, and l 1 , l 2 , . . ., l j is a set of positive or null integers.In the algebraic area enumeration for square lattice walks these integers are the parts in the compositions of the integer n, i.e., n = l 1 + . . .+ l j and all l i positive, with n fixing the length of the walk.But we will consider more general lattice walks where some of the l i 's can be null, in a way to be specified below.
In [1] the focus was on the spectral function which encodes the Hofstadter dynamics.The algebraic area enumeration was obtained in part thanks to an explicit rewriting of the trigonometric sum (1), when evaluated for the Hofstadter spectral function (2), in terms of the binomial multiple sums 1 q q k=1 b p/q l 1 (k)b p/q l 2 (k + 1) . . .b p/q l j (k + j − 1) = ∞ A=−∞ A even e iπAp/q (3) 3) is valid for any set of positive or null integers l i with an A-summation range finite due to the first two binomials, where A appears.In the specific case where the l i 's are all positive -as is the case for the square lattice walks algebraic area enumeration-A is restricted in the interval [−2⌊(l 1 + . . .+ l j )2 /4⌋, 2⌊(l 1 + . . .+ l j ) 2 /4⌋ ].When some of the l i 's are null these bounds can be generalized (see, e.g., the bounds in eq. ( 11)).
We note that when we replace e iπAp/q by 1 in (3) we get the binomial identity 2(l 1 + . . .+ l j ) where the resulting binomial in the LHS1 will be interpreted later on as a factor contributing to the counting of lattice walks.Again, formula (4) is valid for any set of positive or null integers l i ; if the l i 's are all positive the bounds on A are as specified above.
We remark here that the trigonometric sum (1) reduces to the binomial multiple sum given in (3) in the case b p/q (k) = 2 sin(πkp/q) 2 only when l 1 + . . .+ l j < q, i.e., for large enough values of q.In view of the algebraic area enumeration of square lattice walks, where the algebraic area counting ends up being the coefficient of e iπAp/q (see (9) below), this constraint on q eliminates open walks which could be confused with closed ones by periodocity 2 .
In [3] we revisited the algebraic area enumeration of [1] and noted that it admits a statistical mechanical interpretation in terms of particles obeying generalized exclusion statistics [4] with exclusion parameter g = 2 (g = 0 for bosons, g = 1 for fermions, higher g means a stronger exclusion beyond Fermi).Other lattice walks admit a similar interpretation with higher integer values of g.We also introduced the notion of g-compositions where some zeros can be inserted at will inside the set of the l i 's with the restriction that no more than g − 2 zeros lay in succession.The integer n admits g n−1 such compositions.In particular, g = 1-exclusion refers to the unique composition n = n, whereas g = 2-exclusion corresponds to the standard compositions with no zeros at all.We also constructed triangular lattice chiral walks realizing g = 3-exclusion with spectral function b p/q (k) = 2 sin(2πkp/q) 2 sin(2π(k + 1)p/q) (5) We finally hinted at other walks corresponding to statistics with higher values of the exclusion parameter g and to other spectral functions.However, for the triangular lattice chiral walks, as well as for other cases, an explicit algebraic area enumeration formula was missing due to the lack of binomial expressions analogous to (3) for the triangular spectral function (5).
In the present work we focus on filling this gap by uncovering such expressions for entire classes of trigonometric spectral functions generalizing (2) and (5).Namely, we consider, on the one hand b p/q (k) = 2 sin(πkp/q) r (6) and on the other hand b p/q (k) = 2 sin(πkp/q) 2 sin(π(k + 1)p/q) . . . 2 sin(π(k + r − 1)p/q) (7) where in both instances r can be even or odd.The case r = 2 reproduces3 (2) and ( 5) respectively.We will see that the basic structure of the binomial multiple sum (3) naturally generalizes to these cases.In the Appendix we will also derive the relevant generalization for the spectral function where r is even, yet another possible generalization of (5).
Turning to the algebraic area combinatorics per se, these expressions, as already mentioned, will allow for explicit enumeration formulae analogous to the square lattice walks formula obtained in [1] for g = 2 and the Hofstadter spectral function (2).This requires introducing an appropriate weighting coefficient in the summation over compositions of the integer n.We refer to [1] for detailed explanations of how this procedure unfolds and to [3] for the connection to g-exclusion statistics and the resulting generalizations.With the g-exclusion statistics weighting coefficients [3] we can express the lattice walks algebraic area enumeration for g ≥ 2-exclusion and a general periodic spectral function b p/q (k) by means of the g-cluster As already stressed, (9) yields the algebraic area combinatorics provided that an expression analogous to (3) is known for the specific b p/q (k).Indeed, the summation index A in (3) has to be interpreted in (9) as the algebraic area, and the coefficient multiplying the exponential factor e iπAp/q is the sought for algebraic area counting number.It will, in particular, yield the triangular lattice chiral walk counting described by g = 3-exclusion and spectral function (5).
Finally, we will discuss the unexpected occurrence of Apéry-like numbers in the cluster coefficient (9) evaluated at particular values of p/q for certain g-exclusions and spectral functions.Apéry-like numbers are interesting per se since they are cousins of the celebrated Apéry numbers which allow for a proof of the irrationality of ζ(2) and ζ(3).One key characteristic of these numbers is that they are integer solutions of second order recursion relations.As we will see, some of the ζ(2) Apéry-like numbers fascinatingly emerge in the algebraic enumeration formula (9).
2 Trigonometric sums q k=1 b p/q l 1 (k) b p/q l 2 (k+1) We aim at uncovering explicit binomial multiple sums analogous to (3) for the spectral functions (6) and (7).In fact, the form of (3) is quite robust and suggestive, and allows deducing such generalizations by simple deformations while preserving its overall structure.We stress that, from now on, some l i 's can be null according to the g-composition structure discussed previously, i.e., no more than g − 2 zeros in succession inside the set.The A-summation bounds, when specified, will explicitly depend on the parameter g.
2.1 Square lattice walks generalization: b p/q (k) = 2 sin(πkp/q) r We first list two basic facts: • When q → ∞ one obtains the overall counting (10) so we focus on (l 1 + l 2 + . . .+ l j ) such that r(l 1 + l 2 + . . .+ l j ) be even.It means that for r even any set l 1 , l 2 , . . ., l j is admissible, whereas for r odd the l i 's have to be such that their sum be even.
In the r odd case we expect a binomial multiple sum analogous to (11).To see this in full generality, and to give a full proof of the original formula with even r, let us first recall the Poisson summation formula for any q-periodic function f where f is the Fourier transform of f defined as 5 The overall counting, found by replacing e iπAp/q by 1 is Let us consider the function f (x) = 1 q b p/q l 1 (x)b p/q l 2 (x + 1) . . .b p/q l j (x + j − 1) which is indeed q-periodic due to r(l 1 + l 2 + . . .+ l j ) being always assumed even.We have . . .
As stressed above, r(l 1 . . .+ l j ) is even and thus the sum of the k i is an integer.Further, p and q are coprime.These facts imply that the Kronecker-δ in (13) enforces for some integer t.Now j i=1 k i ≤ r(l 1 + . . .+ l j )/2 and thus, under the condition r(l 1 + . . .+ l j )/2 < q, t is necessarily equal to 0, implying that j i=1 k i = 0 and n = 0. From the Poisson summation formula (12) then we infer 14) is the trading of the original sum over k from 1 to q in the LHS for the integral over k from 0 to q in the RHS, which is valid provided that r(l 1 + . . .+ l j )/2 < q.
We can easily check that the trigonometric integral yields the binomial multiple sum (3) in the r = 2 case, or more generally (11) in the r even case.To do so let us proceed from the last line of (13): enforcing the Kronecker δ in the summand we obtain . . .
The change of integration from (1/q) q 0 dk to 1 0 dt in the variable t = kp/q in the second line is justified since r(l 1 +. ..+l j ) is even and the integrand has period 1 in t.We still need to enforce the constraint j i=1 k i = 0 in the summation variables k i .To reproduce the Aexpansion with exponential factors e iπAp/q in the binomial multiple sums ( 3) and (11), we denote by A the coefficient 2 j i=1 (i−1)k i of iπp/q appearing in the exponential of the last line in (15).The resulting system of two equations, j i=1 k i = 0 and A = 2 j i=1 (i − 1)k i , can be readily solved for, e.g., the first two variables k 1 and k 2 , to yield Finally, changing summation variables from k i to −k i and noting that each binomial is invariant under changing the sign of k i , we obtain . . .
rl i rl i /2 + k i i.e., precisely (11) but now valid for r even and r odd, with a specific A-summation dictated by the condition that in (16) the first two binomial entries rl 1 /2 + A/2 + j i=3 (i − 2)k i and rl 2 /2 − A/2 − j i=3 (i − 1)k i still take integer values for all k i ∈ [−rl i /2, rl i /2], i = 3, . . ., j, as was the case in (15) for the first two binomial entries rl 1 /2 + k 1 and It follows that in the case r even, where the k i 's are all integers, A has to be even, and in the case r odd, where the k i 's are either integers or half integers, l 1 + l 2 + . . .+ l j has to be even and A of the same parity as l 1 + l 3 + . . .(or l 2 + l 4 + . ..).In both cases this boils down to A ∈ [−(g − 1)r⌊(l 1 + . . .+ l j ) 2 /4⌋, (g − 1)r⌊(l 1 + . . .+ l j ) 2 /4⌋ ] in steps of 2. We also note that, in this and all subsequent formulae, we follow the convention that the sum of all the lower entries in the binomials in (16) be zero, which fixes the form of such expressions among various equivalent parametrizations.
We can express the A-binomial block in (16) in an integral form by augmenting the LHS to the double integral In the multiple sum of the RHS A is constrained as above, depending on r being even or odd.However, the integral in the LHS is valid for all integer values of A, yielding zero for the values that do not appear in the RHS.
2.2 Triangular generalization: b p/q (k) = 2 sin(πkp/q) 2 sin(π(k + 1)p/q) . . . 2 sin(π(k + r − 1)p/q) We can proceed in exactly the same way for triangular-like spectral functions of the type b p/q (k) = 2 sin(πkp/q) 2 sin(π(k + 1)p/q) . . . 2 sin(π(k + r − 1)p/q) .Again • q → ∞ recovers the overall counting as in (10), so we still focus on sets of l i 's such that r(l 1 + l 2 + . . .+ l j ) is even, again ensuring the q-periodicity of the functions at hand • The rewriting of the trigonometric sum as a trigonometric integral follows the same lines as in (13) under the same condition r(l 1 + . . .+ l j )/2 < q since the sole input in this condition is the highest power of e iπkp/q that appears in b p/q (k) given by (7), which happens to be again r 2.2.1 Triangular chiral walks r = 2: b p/q (k) = 2 sin(πkp/q) 2 sin(π(k + 1)p/q) Following the same steps as in 2.1, we can rewrite the trigonometric sum corresponding to b p/q (k) = 2 sin(πkp/q) 2 sin(π(k + 1)p/q) as the simple integral provided that l 1 + . . .+ l j < q.
] increasing by steps of 2, which in particular implies that A is of the same parity as l 1 + l 2 + . . .+ l j .
Likewise one obtains the binomial multiple sum7 where A has to be of the same parity as l 1 + l 3 + . . .(or l 2 + l 4 + . ..) and obviously a finite range.The cases r = 4 and beyond are treated in the Appendix.
3 Algebraic area enumeration and Apéry-like numbers
with overall counting, given by replacing e iπAp/q by 1 gn n rn rn/2 The second binomial in (22), as initially discussed in (4) and displayed in the various overall counting cases of subsection (2.1), results from the trigonometric sums replacing e iπAp/q by 1 in the limit q → ∞, whereas the first one results from the summation of the exclusion weight coefficients c g over all g-compositions of the integer n.
where u and v respectively stand for the right and up hopping operators on the lattice, with commutation vu = q uv, where q = e iΦ = e i2πp/q is the noncommutativity parameter encoding the presence of the magnetic field perpendicular to the lattice, with Φ the magnetic flux per plaquette.We recover the Hofstadter spectral function as The Hamiltonian describes a random walk with elementary steps up, right followed by up, down, and down followed by left.It means that starting from the origin (0, 0) it reaches after one step the lattice points (0, 1), (1, 1), (0, −1) or (−1, −1) with equal probability.This generates deformed walks on the square lattice (see Fig. 1) which are equivalent through a modular transformation to the usual square lattice walks.(This modular transformation amounts to the transformation u → −uv, which leaves the u, v commutation relation unchanged and turns 21) then yields the desired algebraic area counting [1] b A even e iπAp/q C 2n (A) where counts the number of closed square lattice walks of length 2n -there are overall 2n n 2 of them, see (22)-enclosing an algebraic area A/2 in the interval8 [−⌊n 2 /4⌋, ⌊n 2 /4⌋ ]: indeed the mapping of random walk algebraic area to the Hofstadter model [1] is via the weighting factor q algebraic area , where q = e 2iπp/q , so here, with e iπAp/q appearing in (21), the algebraic area is A/2.

Square lattice walks: b
Let us now look at square lattice walks with g = 2 and r = 4 which are defined in terms of the Hamiltonian The corresponding spectral function b p/q (k) = (q k + q −k ) 4 = 2 cos(2πkp/q) 4 can be put in the standard form (6) for r = 4 by redefining u → iu and q → √ q, which does not affect the counting of walks nor the area weighting.
The Hamiltonian (24) describes a random walk with elementary steps in groups of one random step up or down and two independent random steps right or left.It means that starting from the origin (0, 0) it reaches after one step the lattice points (2, 1), (−2, 1), (2, −1) or (−2, −1) with probability 1/8, or the lattice points (1, 0) or (−1, 0) with probability 1/4.The same walk can be described as a particle hopping on an even or odd square sublattice, where even points are those with x and y coordinates adding to an even integer, the remaining being odd.The walk proceeds randomly on one of the sublattices but at each step it has the option to move to the nearest up or down point of the opposite sublattice, with each such jump contributing a factor of two in the weight of the walk.The Hamiltonian (24) counts the weighted number of such closed walks of a given total area.There are 2n n 4n 2n such closed walks of length 2n, as in (22).The enumeration of such walks enclosing a given algebraic area, with the proper weight, is given by (21): A even where counts the number of closed square lattice walks described above of length 2n and enclosing an algebraic area A/2.
3.1.3Square lattice walks: b p/q (k) = 2 sin(πkp/q) 2 sin(π(k + 1)p/q) 2 Now consider square lattice walks with g = 2 and r = 4 defined by the Hamiltonian The spectral function can be brought to the standard form (8) for r = 4 by an appropriate redefinition of u → −iu b p/q (k) = 2 sin(πkp/q) 2 2 sin(π(k + 1)p/q) Its treatment is given in the subsection 5.2 of the Appendix.
This walk proceeds with sets of one step left or right, one step up or down and another step left or right.With an appropriate redefinition of u and v (modular transformation) this walk can also be mapped to a walk proceeding on odd or even square sublattices, as in the last subsection, but now the weight of jumping on the opposite sublattice is not 2, as before, but rather Q + Q −1 .So in this description the weight of the walks depends explicitly on Q, unlike any other walk we encountered before.
There are again 2n n 4n 2n such closed walks of length 2n.The enumeration of such walks enclosing a given algebraic area, with the proper weight, is given by b where . . .
C ′′ 2n (A) counts again the weighted number of closed square lattice walks described above of length 2n enclosing an algebraic area A/2.It differs from the corresponding number (25) only in the weighting factor when jumping sublattices.
3.1.4Triangular lattice chiral walks: b p/q (k) = 2 sin(πkp/q) 2 sin(π(k + 1)p/q) From ( 19) for the triangular spectral function (7) with r = 2 and g-exclusion we obtain A same parity as n e iπAp/q l 1 ,l 2 ,...,l j g-composition of n c g (l 1 , l 2 , . . ., l j ) with overall counting given by replacing e iπAp/q by 1 gn n 2n n Triangular g = 3 lattice chiral walks correspond to the quantum Hamiltonian with spectral function b p/q (k) = 2 sin 2πpk q 2 sin 2πp(k + 1) q as already given in (5).They are depicted in Figs.2-4 (see [3] for more details; these walks are the generalization to four quadrants of the Kreweras walks [5]).Since the exclusion parameter is g = 3 the counting above reduces to 3n n, n, n which is the number of closed triangular lattice chiral walks of length 3n.The cluster coefficient (27) then yields the triangular lattice chiral walks algebraic area counting where with A in the interval [−n 2 , n 2 ] with same parity as n.
C 3n (A) counts the number of closed triangular lattice chiral walks of length 3n enclosing an algebraic area A. Indeed, the mapping of triangular algebraic area-quantum triangular Hamiltonian discussed in [3] is via q algebraic area where q = e 2iπp/q .Since in b p/q (k) of (7) the building block 2 sin(πkp/q) is used, rather than 2 sin(2πkp/q) as in ( 5), we end up with e iπAp/q in (27) in place of e 2iπAp/q , so that the algebraic area is A. One can directly check by explicit enumeration that when n is odd A is also odd (see, e.g., n = 1 with 3 walks of algebraic area 1 and 3 walks of algebraic area −1) and when n is even A is also even (as in n = 2, with algebraic areas 0, ±2 and ±4).
We conclude our discussion of algebraic area counting by remarking that it was possible to extract explicit expressions in terms of binomial sums for C 2n (A) in (23), C ′ 2n (A) in (25) and C 3n (A) in (28) from the cluster coefficients (21) or (27) because the summation constraints over A in the relevant binomial multiple sums (16) with r = 2, 4 (A even) or (19) with r = 2 (A same parity as n), as well as the summation ranges, depend only on n and not on the l i 's themselves.Similar expressions would apply for walks deriving from odd r binomial sums, like (16) or (20), provided that the binomials appearing in the expressions are understood to vanish for values of A leading to noninteger entries, as discussed after (16).
It is a curious fact that if, in the binomial multiple sums or the cluster coefficients, we sum over all integer values of A without restrictions, and analytically continue the binomials to fractional values using Gamma functions, the resulting infinite sums are closely related to the finite ones over the allowed values of A. This point is detailed and explained in the subsection 5.3 of the Appendix.It means, considering for example the binomial multiple sum (16), that for even r and any set of l i 's, the cumulative sum of the infinite sequence of coefficients of odd A, which are rational numbers times 1/π 2 , converges to the standard binomial counting r(l 1 +l 2 +...+l j ) r(l 1 +l 2 +...+l j )/2 .

Apéry-like numbers
We finally turn to the occurrence of Apéry-like numbers in cluster coefficients (9) when evaluated at certain values of p/q.We stress that we no more view b(n) as generating algebraic area enumerations of actual lattice walks, but instead consider it as a stand-alone mathematical entity that happens to lead to such occurrences.
, 20, 112, 676, 4304, 28496, . . .These are the same Apéry-like numbers as above now occurring for even n's.Indeed, cases r = 2 and (r = 1, n even) are essentially equivalent: calling n = 2n ′ for r = 1, then 2 sin(πkp/q) n=l 1 +l 2 +...+l j with l 1 , l 2 , . . ., l j a composition of n, is in fact (2 sin(πkp/q)) 2 l ′ 1 +l ′ 2 +...+l ′ j =n ′ with l ′ 1 , l ′ 2 , . . ., l ′ j a composition of n ′ , which is the r = 2 result.2 sin πt + π(i − 1)p/q rli 10 n is necesseraly even because l 1 + l 2 + . . .+ l j (which is equal to n) has to be even.The trigonometric identities analyzed in this work, as well as their generalizations to other spectral functions that can be derived along the lines presented here, allow us to obtain expressions for the algebraic area counting of a broad set of random walks on two-dimensional lattices.The only requirement is that these walks be described by a Hamiltonian of the general form introduced in [3], admitting an interpretation as systems of generalized exclusion statistics with specific spectral functions.A wide class of lattice walk models can be embedded into this framework, and we gave a few examples in the present work, most notably the triangular chiral walk introduced originally in [3].
The most obvious and interesting extension of our results would be in obtaining the area counting of other, more general types of walks.From the algebraic point of view, an immediate choice presents itself: the Hamiltonian describes a class of Hofstadter-like models representing generalized random walks on the square lattice, with m = 1 the standard (Hofstadter) random walk and m = 2 the walk studied in subsection 3.1.2.The model for general m represents a walk that proceeds in groups of one random step up or down and then m independent random steps left or right, but other representations are possible by performing modular transformations to the lattice (or redefinitions of the u, v operators in the Hamiltonian).All these walks belong to the class of g = 2 exclusion statistics and their area counting is readily given by the relevant g = 2 cluster coefficients and generalized trigonometric sums.
Clearly this is just the tip of a large iceberg as far as lattice walk models are concerned.For instance, another class of walks at g = 2 would be described by the Hamiltonian This represents walks proceeding with a random step up or down to one of the 2m + 1 neighboring points in the left-right direction of distance up to m from the original horizontal position with equal probability.Again, the combinatorics of these walks are readily obtained with our methods.Yet other walks can be constructed, with asymmetrical propagation rules and belonging to higher g statistics.The only limitation, or criterion, is the potential relevance and physical significance of these walks, and this remains an open field of investigation.
The emergence of Apéry-like numbers within the mathematical structure of these walks is another intriguing but obscure issue.At the present level of our understanding this is something of a mystery, or curiosity.It would be satisfying to have a better understanding of the relation between random walks and Apéry numbers, with an eye to possible applications in the mathematics of ζ-functions and/or statistical models.
Finally, the Hamiltonians H m and Hm presented above are all Hermitian and thus have a real spectrum, generalizing the corresponding spectrum of the Hofstadter model that leads to the celebrated "butterfly" fractal structure.It is expected that the spectrum of all the above models will have a similarly fractal structure.The shape and eigenvalue statistics of the spectrum of these generalized models is an intriguing topic for further research.
One notes that as in previous cases the binomial multiple sum (29) is nothing but the trigonometric integral under the provision that 2(l 1 + . . .+ l j ) < q.

Another triangular chiral walks generalization:
b p/q (k) = 2 sin(πkp/q) r/2 2 sin(π(k + 1)p/q) r/2 with r even When b p/q (k) = 2 sin(πkp/q) 2 we have seen that (3), rewritten as r and r is even to In the r even case we already know that the A even summation in (16) has a finite range and yields exactly the overall integer counting binomial.The A odd summation happens to yield again the same overall binomial but with each term in the sum a rational number times 1/π 2 and an infinite summation range.The 1/π 2 factor comes from the first two binomials in (16) due the relaxation of the constraint that their entries be integers (since A is now odd).Likewise in the r odd case, when l 1 + l 2 + . . .+ l j is even, we already know that A even or odd summations, depending on the parity of l 1 + l 3 + . .., have a finite range and yield the usual overall integer counting binomial; it is still true that summing over A even with l 1 + l 3 + . . .odd or on A odd with l 1 + l 3 + . . .even would yield the same overall counting binomial with again terms 1/π 2 times rational numbers and an infinite summation range.Finally when both r and l 1 + l 2 + . . .+ l j are odd, A even and odd summations have finite range to yield the overall binomial which is in this case 1/π times a rational number.In all these instances the coefficients sum up to r(l 1 +l 2 +...+l j ) r(l 1 +l 2 +...+l j )/2 for both A even or odd summations, with finite or infinite ranges depending on the situation.
To better understand these weird A-summations, let us first focus on the regular Asummations and consider the LHS of (17) i.e., the binomial multiple sum rl i rl i /2 + k i One wishes to go backward and get the double integral in the RHS of (17), which, when summed over A, directly yield the overall counting binomial r(l 1 + l 2 + . . .+ l j ) r(l 1 + l 2 + . . .+ l j )/2 For simplicity let us consider the case r even: since r is even, all the k i 's i = 3, . . ., j are integers, and since we know that A has then to be even (see below (16)), in the first two binomials both rl 1/2 + A/2 + j i=3 (i − 2)k i and rl 2 /2 − A/2 − j i=3 (i − 1)k i are integers.Using that for an integer n dt ′ e iπA(t ′ −t) j i=1 rl i rl i /2 + k i e 2iπk i (i−1)t ′ −(i−2)t) i.e., since obviously rl i /2 k i =−rl i /2 k i integer rl i rl i /2 + k i e 2iπk i (i−1)t ′ −(i−2)t = 2 cos π ((i − 1)t ′ − (i − 2)t) dt"e iπAt" j i=1 2 cos π ((i − 1)t" + t) We have to sum over A even: since A even e iπAt" = ∞ n=−∞ δ(t", n) A even where the overall binomial counting has been obtained as expected.
12 Or equivalently as in the RHS of (17)

p j i=1 k i = qn and thus j i=1 k
i = tq and n = tp