Differentiation identities for hypergeometric functions

It is well-known that differentiation of hypergeometric function multiplied by a certain power function yields another hypergeometric function with a different set of parameters. Such differentiation identities for hypergeometric functions have been used widely in various fields of applied mathematics and natural sciences. In this expository note, we provide a simple proof of the differentiation identities, which is based only on the definition of the coefficients for the power series expansion of the hypergeometric functions.


Introduction
The theory of hypergeometric functions is fundamental in mathematical physics since they contain as special cases almost all the special functions commonly used for centuries. Among the family of hypergeometric functions, the most well-known ones are Gauss hypergeometric function and Kummer confluent hypergeometric function, and their natural extension is known as generalized hypergeometric function [1,2]. A lot of efforts have been made to deepen understanding of the hypergeometric functions, which have accumulated hundreds of useful formulae [3][4][5][6], such as relation to other special functions, series expansions, differential properties, contiguous relations, variable transformations, integral representations etc. These formulae clarify the nature of the hypergeometric functions from various aspects, and have been widely used as powerful tools in diverse research fields [7].
Among such formulae, let us focus on the differentiation identity for the hypergeometric functions. Consider the Gauss hypergeometric function as an example, which is defined by an infinite series where (a) k is the Pochhammer symbol defined by (a) k = Γ(a + k) Γ(a) = 1, (k = 0), a(a + 1) · · · (a + k − 1), (k > 0).
Here, it is assumed that c is not a nonpositive integer to avoid the singular behavior of the coefficients in (1). The series converges on |z| < 1. Several differentiation identities are wellknown in the literature. One of the famous identities is as follows (see, e.g., [4,Eq. (15.2.4)]): While it is typically not clearly mentioned in the literature, this identity does not hold if c ∈ Z <n+1 . For c ∈ Z <n+1 , the right-hand side is singular. For c ∈ Z ≤0 , the left-hand side is also singular. However, for c = 1, 2, · · · , n, the left-hand side is not singular, despite that the right-hand side is singular. It is thus natural to ask whether there exists a different kind Table 1. Numerical evaluation of the left-hand side (f L ) and the first (f R1 ) and second (f R2 ) lines of the right-hand side of (4) for several values of c with a specific parameter set n = 4, a = 1/2, b = 2/3, and z = 1/3.
; z , (c = 1, 2, · · · , n + 1). (4) Note that the first and second lines of the right-hand side are identical for c = n + 1. In Table 1, we present example numerical values of the left-hand side (f L ) and the first (f R1 ) and second (f R2 ) lines of the right-hand side of (4) for several values of c, while we fix the other parameters to a specific parameter set n = 4, a = 1/2, b = 2/3, and z = 1/3. They match each other for the range of the validity of the identity (4). As explained above, f L is regular but f R1 is singular for c = 1, 2, 3, 4, and f R2 precisely complements the identity for the exceptional case. On the other hand, for c = 6, 7, · · · , f R1 is regular but f R2 is singular since the lower parameter n − c + 2 in f R2 is a nonpositive integer. In Fig. 1, we depict the left-hand side (f L , black) and the first (f R1 , green) and second (f R2 , red) lines of the right-hand side of (4) as a function of the parameter c, with the same parameter set. The f R1 and f R2 intersect each other at c = 5. For c = 1, 2, 3, 4 and c = 6, 7, · · · , one of f R1 and f R2 is singular, but we see an interesting behavior where the regular one complements the singular one. Therefore, the difference f R1 − f R2 has a peculiar periodic behavior as a function of c.
Of course, one can derive the formula for the case c = 1, 2, · · · , n by applying the analytic continuation to the right-hand side of (3). It is then possible to enlarge the validity of the formula and to obtain interesting pairs of functions demonstrated above. However, such a derivation is based on the assumption that one already knows the original formula (3). Thus, this strategy may not work for more general functions, if no differentiation identities are known for those functions.
The aim of this expository note is to provide a simple derivation of the differentiation identities for the generalized hypergeometric function, treating the well-known cases such as (3) and lesser-known exceptional cases on an equal footing. The advantage of this derivation is that, unlike the approach relying on the analytic continuation mentioned above, it is based  Figure 1. The left-hand side (f L , black) and the first (f R1 , green) and second (f R2 , red) lines of the right-hand side of (4) for a specific parameter set n = 4, a = 1/2, b = 2/3, and z = 1/3.
only on the definition of the coefficients for the power series expansion of the hypergeometric functions and does not require a priori knowledge of the differentiation identities. The strategy presented here may work for more general functions [8], while the coefficients for the power series expansion are in general not so simple.
In the rest of the paper, we derive differentiation identities for the generalized hypergeometric function, including the hypergeometric function and confluent hypergeometric function as special cases, in a concise and systematic way to exhaust all the possible identities, some of which are omitted in the literature.

Prelude
In this section we summarize preliminary definitions and notations. Generalized hypergeometric function is defined formally by an infinite series with coefficients with the Pochhammer symbol (a) k defined in (2). Here, p, q ∈ N, and other parameters and the variable z are complex numbers in general.
The case with (p, q) = (2, 1) yields the hypergeometric function, whereas the case with (p, q) = (1, 1) yields the confluent hypergeometric function. For these cases, the notations ; z are also commonly used in the literature. These two hypergeometric functions are especially important and have been extensively studied since they are exact solutions for second-order differential equations known as (confluent) hypergeometric equations, which encompass a broad class of differential equations appearing in mathematical physics. For simplicity, we use the notation a = (a 1 , · · · , a p ) and b = (b 1 , · · · , b q ) and express the generalized hypergeometric function (5) as p F q a b ; z . We also adopt the notation . Further, we write the product of the Pochhammer symbols as (a) k = (a 1 ) k · · · (a p ) k and (b) The coefficient (6) is then written as ; z remains the same under any permutation of the arguments of a, and any permutation of the arguments of b. Therefore, for the following, when we focus on an argument of a or b, we always take the first argument a 1 or b 1 without loss of generality, and write the rest arguments asâ = (a 2 , a 3 , · · · , a p ) orb = (b 2 , b 3 , · · · , b q ). Note thatâ is empty for p = 1, andb is empty for q = 1. The radius of convergence of the generalized hypergeometric series (5) is as follows [1,2]: If p < q + 1, the series converges for all values of z on the complex plane, provided that the coefficients (6) are not singular. If p > q + 1, the series does not converge except for z = 0, and hence the function is defined only when the series terminates. If p = q + 1, the series converges when |z| < 1. Whether it converges for the case p = q + 1 when |z| = 1 is more subtle. It converges when z = 1 if and If none of the upper parameters a is a nonpositive integer, we assume that none of the lower parameters b is a nonpositive integer to avoid the possibility of singular coefficients. On the other hand, if one or more of the upper parameters a are nonpositive integer, the series (5) terminates and p F q a b ; z becomes a polynomial. For instance, if a 1 = −m such that m ∈ Z ≥0 , the coefficient c k vanishes for k ≥ m + 1, and hence p F q a b ; z is at most m-th order polynomial. Namely, we can write down the generalized hypergeometric function as a finite series as This expression is also valid even This definition is a natural generalization of the one for the Gauss hypergeometric function, which is commonly used in the literature, see, e.g., [6, Eq. (15.2.5)] or [9, page 39]. Throughout the present paper, we adopt the definition (10). In other words, for the case of (9), we relax the assumption on the lower parameters b as follows: None of the lower parameters b is a negative integer −k such that k ∈ Z <m . By definition, it holds that if c = d. For brevity, for the following we call this reduction procedure "cancellation" between c and d.

Main Theorem
Theorem 1. For n ∈ N and r ∈ C, the following differentiation identities for the generalized hypergeometric function p F q hold: (r = 0, 1, · · · , n), We can easily check that the identities in Theorem 1 (and Corollaries 1.1 and 1.2 below) hold numerically along the same line of Table 1 and Fig. 1, while we do not repeat them here.
Note that, similarly to the case c = n + 1 for (4), there are some cases where the formulae are identical; The first and second lines of the right-hand side of (12) with r = n, (15) with b 1 = n + 1, and (16) with r = 0 are identical, respectively. Note also that, while different kinds of the generalized hypergeometric functions, p F q and p+1 F q+1 , appear on both sides of the above identities, as expected, the term-by-term differentiation of a convergent power series has the same radius of convergence as the original power series. Indeed, both radii of convergence of p F q and p+1 F q+1 are determined by the value of p − q = (p + 1) − (q + 1), as explained in §2.
While this type of differentiation identities can be found in the literature, some of them are not written down explicitly. Specifically, the first line of (12), (13), (14), and the first line of (15) correspond to, e.g., Eqs.  [4]), but the other identities, i.e, (12), the second line of (15), and (16) seem to be omitted.
As we discussed in §1, one can formulate the identities for such exceptional cases by applying analytic continuation to the identities for the well-known cases. However, such a derivation assumes that one already knows the identities for some cases. Thus, this strategy may not work for more general functions, if no differentiation identities are known for those functions. From this point of view, we prove Theorem 1, treating all the identities on an equal footing, without assuming a priori knowledge of some part of the identities. We provide a proof of Theorem 1 in §4, which is fairly concise and is based only on the definition of the coefficients for the power series expansion of the hypergeometric functions.
More generally, one can also apply generalized Kummer-type transformations for p+1 F p [10] and p F p [11] to generate similar identities.

Proof of Theorem 1
We consider a differentiation d n dz n z r p F q According to which terms in the series survive after performing the term-by-term differentiation in the right-hand side, we consider the following three cases separately: (1) r ∈ Z <n , (2) r = 0, 1, · · · , n, (3) r ∈ Z <0 .
Note that the first two cases overlap for r = n, corresponding to the similar overlapping in (12) and (15), for which the formulae coincide. Below we prove the identity (12) for each case, from which we derive other identities (13)-(16) by using the reduction (11).
4.1. r ∈ Z <n . In this case, all the terms in the series in the right-hand side of (36) survive after performing the term-by-term differentiation. We thus obtain d n dz n z k+r = (k + r)(k + r − 1) · · · (k + r − n + 1)z k+r−n = (k + r − n + 1) n z k+r−n .
Using the definitions (5) and (6), and an identity (a + k) n (a) n = (a + n) k (a) k , hypergeometric function in the right-hand side of (46) does not exist. Consequently, we obtain d n dz n z −m p F q a b ; z = (−m − n + 1) n z −m−n p F q −m + 1,â b ; z , which, after substituting m = −a 1 − n + 1, yields (14) for the case a 1 ∈ Z <0 . Finally, we consider a cancellation in the second term in the right-hand side of (46). Again, the cancellation between n + 1 and b 1 + m + n is impossible in general to avoid the singular behavior of, in this case, the left-hand side. The cancellation between m + n + 1 and a 1 + m + n occurs if a 1 = 1. In this case, (46) yields the second line of (16).