Hypergeometric Functions at Unit Argument: Simple Derivation of Old and New Identities

The main goal of this paper is to derive a number of identities for the generalized hypergeometric function evaluated at unity and for certain terminating multivariate hypergeometric functions from the symmetries and other properties of Meijer's $G$ function. For instance, we recover two- and three-term Thomae relations for ${}_3F_2$, give two- and three-term transformations for ${}_4F_3$ with one unit shift and ${}_5F_4$ with two unit shifts in the parameters, establish multi-term identities for general ${}_{p}F_{p-1}$ and several transformations for terminating Kamp\'e de F\'eriet and Srivastava $F^{(3)}$ functions. We further present a presumably new formula for analytic continuation of ${}_pF_{p-1}(1)$ in parameters and reveal somewhat unexpected connections between the generalized hypergeometric functions and the generalized and ordinary Bernoulli polynomials. Finally, we exploit some recent duality relations for the generalized hypergeometric and $q$-hypergeometric functions to derive multi-term relations for terminating series.


Introduction and preliminaries
Here and throughout the paper we will use the standard symbol p F q (a; b; z) for the generalized hypergeometric function with parameter vectors a ∈ C p , b ∈ C q \{0, −1, . . . }, see [2, Section 2.1], [24,Section 5.1], [32, for precise definitions and details. We will omit the argument z = 1 from the above notation throughout the paper. The guiding idea of this work is to employ the properties of Meijer's G function G m,n p,q to discover new identities for the generalized hypergeometric function r+1 F r . This idea proved very fruitful and appears as a recurrent theme in a series of papers published by the second and the third named authors over past decade, including [15,16,17,18,19,23]. Let us introduce some notation and definitions. For any vectors a ∈ C n , b ∈ C m and a scalar β define Γ(a) = n i=1 Γ(a i ), a + β = (a 1 + β, . . . , a n + β), a [k] = (a 1 , . . . , a k−1 , a k+1 , . . . , a n ), where Γ(·) is Euler's gamma function. Given integers 0 ≤ n ≤ p, 0 ≤ m ≤ q and complex vectors a ∈ C n , b ∈ C m , c ∈ C p−n , d ∈ C q−m , such that a i − b j / ∈ N for all i, j, Meijer's G function is defined by the Mellin-Barnes integral of the form G m,n p,q z a, c b, d : (1.2) where the contour L is a simple loop that starts and ends at infinity and separates the poles of s → Γ(b+s) leaving them on the left from those of s → Γ(1 − a−s) leaving them on the right. Details regarding the choice of the contour L and the convergence of the above integral can be found, for instance, in [22,Section 1.1], [33,Section 8.2], [32,Section 16.17] and [16,Appendix]. If some of the vectors a, b, c, d are empty, they will be omitted from the above notation. The integral in (1.2) can be evaluated by the residue theorem which, in the case when all poles of the integrand are simple, leads to a finite sum of hypergeometric functions. This expansion was derived by Meijer himself, see details in [ The above special G p,0 p,p case of Meijer's G functions plays an important role in the solution of the generalized hypergeometric differential equation, see [16,29]. It was studied in great detail by Nørlund in [29] under different notation and without mentioning the G function. As G p,0 p,p (t) = 0 for t > 1 according to [15,Property 3], the Mellin transform of G p,0 p,p (t) reduces to the integral: Γ(a + s) Γ(b + s) = This representation turned out to be very useful in studying the properties of the function p+1 F p [15,16,17,19]. In this paper we will combine it with the expansion [15, equation (11)], [23, equation (1.10)] where ν(a; b) is defined in (1.1), found by Nørlund, see [29, equations (1.33), (1.35) and (2.7)]. It converges in the disk |1 − t| < 1 for all complex values of parameters and each ω = 1, 2, . . . , p.
Note that if −ν(a; b) = m ∈ N 0 , then the first m + 1 terms in (1.6) vanish. The coefficients g p n (x; y) are polynomials symmetric with respect to separate permutations of the components of x = (x 1 , . . . , x p−1 ) and y = (y 1 , . . . , y p ). These coefficients serve as the key to many properties of the generalized hypergeometric function and play an important role in this paper. Their known and new properties will be discussed in detail in the following Section 2. In particular, we will present expressions for these coefficients in terms of multiple hypergeometric series and in terms of generalized Bernoulli polynomials. The obvious symmetries of these coefficients lead to various identities for the terminating univariate and multivariate hypergeometric series. These facts will be summarized in Theorem 2.1 (transformations of the multiple hypergeometric series) and Theorem 2.13 (relation to the generalized Bernoulli and the complete Bell polynomials) in Section 2 and further exemplified in Examples 2.2 through 2.17 in the same section. The rest of the paper is organized as follows. In Section 3 we give an elementary derivation of the known expansion (3.1) of the function z → p+1 F p (a; b; z) near z = 1 and derive a presumably new formula for the analytic continuation of (a, b) → p+1 F p (a; b; 1) as a function of parameters presented in Theorem 3.1. We further explore some consequences of these formulas including two-and three-term Thomae's relations, two-and three-term transformations for 4 F 3 with one unit shift. We further pursue this topic in Section 4, where we focus on transformations for 5 F 4 (1) with two unit shifts in parameters. The main result of this section is Theorem 4.1 containing a two-term transformation for such type of 5 F 4 (1) series. In Section 5 we demonstrate that the known multi-term transformations for p+1 F p (1) (5.1) and (5.2) (which can be viewed as far-reaching generalizations of the three-term Thomae's relations) are straightforward consequences of the properties of the G function. We further present another multi-term identity in Theorem 5.1 which completes a calculation contained in the classical monograph by Slater [37]. The final Section 6 is devoted to transformations of the terminating series. Here, we give two identities for the terminating generalized hypergeometric series derived from the so-called duality relations found recently in [14]. These results are presented in Theorems 6.2 and 6.3. Furthermore, we employ some recent q-series identities from [12] to deduce three-term relations for rather general terminating hypergeometric series given in Theorems 6.6 and 6.7.
Introduce the new summation indices according to Then, applying the relations 3) can be written as (p − 2)-fold terminating hypergeometric series where l s = (l 1 , . . . , l s ), |l s | = l 1 + · · · + l s , valid for p ≥ 3. We were unable to find any name attached to this particular type of series in the generality given above. Nevertheless, an anonymous referee drew our attention to an identity for q-hypergeometric series due to George Andrews contained in [1,Theorem 4]. By taking the limit q → 1 and appropriately changing notation, his result leads to the following relation: where as before x ∈ C p−1 , y ∈ C p , l s = (l 1 , . . . , l s ), |l s | = l 1 + · · · + l s and µ p = y p + y p−1 for brevity. This surprising formula expresses a multiple series somewhat similar to (2.6) in terms of very-well poised 2p+1 F 2p (1). Nevertheless, to the best of our understanding the left hand side of Andrew's identity is essentially different from the right-hand side of (2.6). To justify this claim let us compare them for p = 3. Then, the left-hand side of (2.7) takes the form (recall that omitted argument equals 1): while formula (2.6) reads where ν 2 is defined in (2.4). The first expression represents the general Saalschützian 4 F 3 , while the second is the general terminating 3 F 2 . As there is no (known) transformation mapping one into the other, these two expressions are likely to be essentially different.
For p = 4, i.e., x = (x 1 , x 2 , x 3 ), y = (y 1 , y 2 , y 3 , y 4 ), we have by (2.1) and in view of (2.5), Alternatively, (2.6) yields where a ∈ C p , b ∈ C s , c ∈ C q , d ∈ C t , e ∈ C r , f ∈ C u and z, w can take arbitrary values as long as the above series terminates. In the general case, convergence conditions are given in [13,Chapter 14], while the analytic continuation can be constructed following the ideas from [4]. For p = 5, i.e., x ∈ C p−1 , y ∈ C p , by (2.1) we obtain which, according to (2.6), is equal to with parameter vectors of arbitrary finite sizes and arbitrary values of variables as long as the above series terminates. Another recurrence relation found by certain renaming in Nørlund's formula [29, formula (1.41)] is given by with p ≥ 3 and initial values (2.2). Using this recurrence for p = 3 we get (see also [15,Property 6]): For p = 4, denoting ψ m = m j=1 y j − m−1 j=1 x j = ν m + x m we have (see also [15,Property 6]): The recurrence (2.11) can also be solved in general. For odd p ≥ 3 we obtain where j 0 = 0, j p−1 = n. Similarly, for even p ≥ 4, we get Introduce the new summation indices according to Then, applying the relations (2.5) for odd p ≥ 3 we get the multiple hypergeometric representations (ψ m = ν m + x m ): . . , l m ) and |l m | = l 1 + · · · + l m . In a similar fashion, for even p ≥ 4 we get For p = 4 this yields (returning to ν m = ψ m − x m ): For p = 5 we get the following expression in terms of Srivastava's F (3) [39, Section 1.5, formula (14)]: Another symmetry of the Nørlund's coefficients comes from the observation mentioned above: where ω ∈ {1, . . . , p}. In particular, putting t = 1 we get g p m+1 a [ω] ; b on the right-hand side which implies that The main findings of this section so far can be summarized as follows.
Theorem 2.1. The terminating multiple hypergeometric series defined in (2.6) and (2.14) are equal, symmetric with respect to permutations of the components of x, symmetric with respect to permutations of the components of y, satisfy the transformation formula (2.18) and are nonnegative if ν m ≥ 0 and y m+1 ≥ x m for m = 1, . . . , p − 1 or, equivalently, if ν m ≥ 0 and ν m − y m + y m+1 ≥ 0 for m = 1, . . . , p − 1.
Example 2.7. The symmetry from the previous example applied to (2.15) yields Note that for γ, δ, µ ≥ 0, η ≥ α ≥ 0 and β ≤ µ both sides of the above formula have the same sign as Example 2.8. If we take p = 3 in (2.18) and use (2.12) for g 3 n we again arrive at the second relation in Example 2 equivalent to [38, Appendix, formula (III)].
Example 2.11. The equality right-hand side of (2.9) = right-hand side of (2.15) represents a presumably new transformation for the terminating Kampé de Fériet function: The condition α, β, γ, µ, η, δ ≥ 0 ensures that both sides of the above formula are non-negative.
Let us remark in passing that a number of transformations for the terminating hypergeometric and Kampé de Fériet functions were motivated by the study of the symmetries of the 3 − j, 6 − j and 9 − j coefficient appearing in the quantum mechanical treatment of angular momentum, see [38,40] and references therein for details.
Next, recall that the Bernoulli-Nørlund (or the generalized Bernoulli) polynomial B In particular, B is the classical Bernoulli polynomial. In [16, Theorem 3] we found the following alternative representation for Nørlund's coefficients: The recurrence (2.22) can be solved giving the following explicit expression for l r (x; y; α) [34, formula (4.1)] (we use the abbreviated notation q m := q m (x; y; α)): Another way to write this formula is to invoke the complete exponential Bell polynomials Y r generated by [8, formula (11.9)] and written explicitly as [8, formula (11.1)] These polynomials can also be found using a determinantal expression, see [9, p. 203]. Comparing the above explicit formulas for l r (a; b; α) and Y r we conclude that whereq m = (m − 1)!q m with q m from (2.23). Substituting this into (2.21) we obtain the second main result of this section.
Theorem 2.13. For arbitrary α the following identity is true:
Remark 2.14. Formula (2.24) gives single sum expression for the coefficients g p n (x; y) manifestly symmetric in the components of x and y. The computational complexity of this formula does not increase with growing p unlike (2.6) and (2.14). Particular cases obtained by expressing g p n using (2.8), (2.12), (2.9), (2.15), (2.10) or (2.16) lead to rather exotic identities connecting terminating 3 F 2 , Kampé de Fériet and Srivastava F (3) functions with the complete exponential Bell and the Bernoulli-Nørlund polynomials.
Combining with Example 2.12 we can write a similar expression for another version of F (3) .
We conclude this section with the remark that explicit expressions for g p 1 and g p 2 for any p can be found in [

Function p+1 F p at and near the singular unity revisited
The following expansion for p+1 F p (z) around the singular point z = 1 was first obtained by Nørlund [29] with further contributions by Olsson [31], Bühring [6], Saigo and Srivastava [35]: where ν = ν(a; b) is assumed to be non-integer. Various expressions are known for the coefficients f p (k; a; b) and h p (k; a; b), see [ where ν [1,2] = b 1 + · · · + b p − a 3 − · · · − a p , and g p n (·) are Nørlund's coefficients defined in (1.6). Next, applying formula [2, formula (2.3.13)] connecting 2 F 1 (z) and 2 F 1 (1 − z) in the case of non-integer parametric excess (equal to ν + n here) and rearranging slightly we arrive at Using the recurrence relation (2.11) we obtain the formulas for the coefficients f p (k; a; b) and h p (k; a; b) from (3.1) in terms of Nørlund's coefficients:  [20]. Let us also remark that a similar substitution of expansion (1.6) into the Laplace transform representation 2)] via multiplication by e −z and change of variable u = 1 − t, leads immediately to asymptotic expansion of p F p (z) as z → ∞ recovering the main result of [41]. Further development of these ideas can be found in [23]. Next, we note that for Re(ν(a; b)) > 0 by setting z = 1 in expansion (3.1), we get Hence, omitting the unit argument from the notation of p+1 F p , we arrive at Γ(ν + n)g n a [1,2] ; b Γ(ν + a 1 + n)Γ(ν + a 2 + n) .
It follows from the estimate in [23, Lemma 1.1] that the series on the right-hand side converges if Re a [1,2] > 0, while the series on the left hand side converges if Re(ν(a; b)) > 0, so that the expression on the right gives the analytic continuation in parameters for p+1 F p (a; b; 1). Formula (3.4) in a slightly different form was obtained by Nørlund [29, formula (4.6)]. It is also equivalent to [6, Theorem 1], modulo simple transformation of coefficients in Bühring's formula. We further remark that although for Re(ν(a; b))) < 0 the hypergeometric series on the left hand side of (3.4) diverges, in view of representation (1.3) and expansion (2.17) for −ν(a; b) = m ∈ N 0 , we obtain the limit formula where ω ∈ {1, . . . , p} is arbitrary as we mentioned below (2.17).
In [16,Theorem 3.10] by reinterpreting some Nørlund's results, we gave the following expression for the coefficients h p (m; a; b) (after some change of notation): Combining (3.3) with (3.5) and (3.6), we arrive at Theorem 3.1. The following representation holds true
Remark 3.2. If we use (3.7) instead of (3.6) we obtain This series also converges for Re a [1,2] > 0. However, this expansion seems to be less useful than (3.8) as the higher order hypergeometric function appears on the right-hand side.  [1,2] > 0 is violated, then we can use the following straightforward decomposition [15, inequality (31)] Choosing M sufficiently large we can always ensure the condition Re(a [1,2] + M ) > 0 in the second term, while the first term is a rational function. Note, that the parametric excess in the second term on the right-hand side Next, we explore some consequences of (3.8). Taking p = 2 and applying the Chu-Vandermonde identity to the 2 F 1 polynomial on the right, after some simplifications we arrive at where the symbol idem(b 1 ; b 2 ) after an expression means that the preceding expression is repeated with b 1 , b 2 interchanged. This formula is yet another instance of the three-term Thomae relations. It can be obtained by eliminating F n (0) from the pair of equations [3, formulas 3.7(4) and 3.7 (6)]. Before we move forward we need the following lemma.
where λ 1 , . . . , λ m are the zeros of the polynomial (3.10) In particular, for m = 1, Proof . In order to establish (3.9) first note that for integer 0 ≤ k ≤ m: This formula is also true if m > n with the standard convention (a) r = (−1) r /(1 − a) −r for integer r < 0. Applying this relation we get where P m (x) is defined by (3.10). Factoring P m (x) we get: where A denotes the coefficient at x m in P m (x). It remains to note that We now take p = 3 and a 3 = b 3 + m, m ∈ N, in (3.8). The term corresponding to k = 3 vanishes and we are left with two infinite series with terms involving If we apply Sheppard's transformation (2.19) to each of these two functions we obtain (keeping in mind that a 3 = b 3 + m, ν = ν(a; b)): Γ a [1,2] sin(πν) where the vector λ = (λ 1 , . . . , λ m ) comprises the zeros of the polynomial and idem(b 1 ; b 2 ) contains the zeros of the polynomial P m (a 4 ; b 2 , b 1 , b 3 ; x). The simplest and the most interesting particular case is m = 1. In this case we get the following three-term relation for 4 F 3 (1) with one unit shift in the parameters: We now turn our attention to the consequences of (3.4). For p = 2 substitution g n (a; b) = (b 1 − a) n (b 2 − a) n /n! (see (2.2)) gives yet another proof of the two-term Thomae relation [2, Corollary 3.3.6]. For p = 3 substituting (2.12) into (3.4) yields: This formula differs from [6, formulas (2.14) and (2.10)], but can be reduced to it by an application of Whipple's transformation for terminating 3 F 2 (1) [2, p. 142, top]. If we put a 1 = b 1 + 1 in (3.11) we arrive at a two-term transformation for 4 F 3 with one unit shift studied by two of us recently in [21, identity (7)]. Further consequences of (3.4) are explored in the following section.

Transformations of 5 F with two unit shifts
In our recent paper [21] we have studied a group of transformations of 4 F 3 (1) with one unit shift in the parameters. We have shown that this group is generated by two-term Thomae transformations and contiguous relations for 3 F 2 (1). In this section we will demonstrate that a similar group can be generated for 5 F 4 with two unit shifts. Let us note that the study of the summation and transformation formulas for the generalized hypergeometric function with integral parameter differences was initially motivated by problems from mathematical physics, see, for instance, [26,28,36]. The most general linear transformations for this class of hypergeometric series at arbitrary argument were discovered by Miller and Paris [27] (see an alternative derivation in [19]), while quadratic and cubic transformations were found recently by Maier [25].
Theorem 4.1. The following two-term transformation holds: where s = d + e − a − b − c − 2 and γ 1 , γ 2 are the roots of the second degree polynomial Proof . Formula (3.4) applied to the left hand side of (4.
Noting that the quadratic polynomial Substituting the above formula into (4.2), we arrive at (4.1).
Another transformation of a similar flavour as (4.1) has been recently found by us in [18, formula (59)], namely, Transformations (4.1) and (4.3) can be iterated and composed with each other. Together with the obvious invariance with respect to permutations of the upper and the lower parameters they generate a group, which can be shown to be isomorphic to the direct product of two-term Thomae transformations for 3 F 2 with contiguous relations for 3 F 2 . This claim can be verified using the approach from [21]. We further believe that similar transformations hold for higher order p+1 F p with appropriate number of unit shifts in parameters. We envision further investigation of this topic in a future publication.

Multi-term transformations of the non-terminating series
Nørlund proved the following identity [29, relation (5.8)]: where all series involved converge if Re(ν(a; b)) > 0. This identity was later rediscovered by Wimp in [42,Lemma 2]. We note that it also follows immediately on substituting (1.3) into (1.5) and integrating term-wise using the beta integral: On the other hand, formulas (1.3) and (1.6) imply immediately that for p+1 We note that particular cases of identities (5.1) and (5.2) appeared many times in the literature proved by various methods. For instance, the three-term Thomae relation for 3 F 2 [3, formula 3.2(2)] is a particular case of (5.1), while its variation [11, formula (10)] and 4 F 3 generalization [11, formula (19)] reduce to (5.2) after appropriate change of notation. We note also that (5.2) is not a straightforward rewriting of (5.1). Indeed, setting b 1 = 0 brings (5.2) to the form: Changing notation according to 1 − c → a, 1 − b [1] → b we get where all series involved converge if Re(ν(a; b)) > 0. This is manifestly different from (5.1). It remains unclear for us, however, if (5.1) and (5.3) can be obtained from each other by the appropriate compositions. Our next theorem gives another multi-term identity for p+1 F p containing p or more terms and not immediately found in the literature (other than for p = 2, see remark below).
Theorem 5.1. Suppose 0 ≤ n ≤ p, 0 ≤ m ≤ p are integers that satisfy m + n ≥ p and a ∈ C n , b ∈ C m , c ∈ C p−n , d ∈ C p−m are complex vectors that satisfy   exists for − min(Re(b)) < Re(s) < min(1 − Re(a)) under conditions of the theorem. Assuming that s = 0 belongs to this range we can take the Mellin transform with s = 0 on both sides of (5.5) to obtain which proves (5.4a) under the restriction − min(Re(b)) < 0 < min(1 − Re(a)). This restriction can now be removed by analytic continuation. 6 Multi-term transformations of the terminating series Introduce the following notation: m = (m 1 , . . . , m r ) ∈ Z r and n = (n 1 , . . . , n r ) ∈ Z r , M = m 1 + · · · + m r , N = n 1 + · · · + n r , In [14,Theorem 1] A. Kuznetsov and the second author established the following identity for arbitrary a, b ∈ C r such that the components of a are distinct modulo integers, and |z| < 1: The authors did not give any explicit expression for the numbers β j . Our aim here is to employ the above formula for deriving some identities for the generalized hypergeometric function evaluated at 1. To this end we will need the explicit expressions for β j , which we present in the following lemma.
Lemma 6.1. Formula (6.1) holds true for where each term with j + n i < 0 is assumed to equal zero.
Proof . Writing S(z) for the left hand side of (6.1) we get by collecting terms where, according to the formula on page 4 of [14], we have the first equality below: The second equality above is obtained by an application of the easily verifiable identities (z − j) n = (z) n (1 − z) j (1 − z − n) j and (m − j)! = (−1) j m! (−m) j . Hence, Theorem 6.2. For all k ≥ −m min we have α k = q p (k) , or, in detail, where q p is defined in (6.4). In particular, q −1 (k) = 0, q 0 (k) = 1, and, furthermore, Proof . Keeping the notation from the proof of Lemma 6.1 we can write where the first sum is zero if −n max > −m min − 1. Furthermore [14,Lemma 1] shows that α k in the second sum is precisely the polynomial q p (k) defined above, see the first formula below the proof of [14, Lemma 1].
Writing as before ν = ν(a; b) = r i=1 (b i − a i ), we have Theorem 6.3. Let a, b ∈ C r , m, n ∈ Z r and ν − M + N − 1 < 0, −ν + r − 1 < 0. is valid if ν(c; d) = r−1 j=1 d j − r j=1 c j < 0. Hence, under conditions (6.5) the above relation is applicable to both series in each summand in (6.1). Summing both inequalities in (6.5) we see that M − N − r + 1 > −1 which implies by definition of p that p = M − N − r + 1. A straightforward calculation using the asymptotic relation above then shows that the total power of 1 − z does not depend on i and equals −p − 1. Hence, we get from (6.1) as z → 1 In the limit z = 1 we thus obtain after simple rearrangement and application of the reflection formula Γ(x)Γ(1 − x) = π/ sin(πx) that In a rather recent work [12] the authors discovered some curious three-term duality relations for the q-hypergeometric functions. As q → 1 they naturally lead to identities for ordinary generalized hypergeometric functions which do not seem to appear in the literature previously. The general case and its corollary are presented below. We will then use them to derive threeterm relations for terminating hypergeometric series evaluated at ±1.
Proof . Equating coefficients at z n on both sides of (6.7), using the standard Cauchy product and relations (2.5) we arrive at the result.
In a similar fashion from Lemma 6.5 we obtain the following: Theorem 6.7. For a given integer r ≥ 1 suppose a ∈ C r+1 , b ∈ C r . Then for all integer n ≥ 0