Transformation and summation formulas for double hypergeometric series

A number of new transformation formulas for double hypergeometric series are presented. The series appearing here are the so-called Kampé de Fériet functions of type F_1:1;2^0:3;4(1,1) and F_0:2;2^1:2;2(1,1). The transformation formulas relate such double series to a single hypergeometric series of ₄F₃(1) type. By specializing certain parameters, a list of new summation formulas for F_0:2;2^1:2;2(1,1) series is obtained. The origin of the results comes from studying symmetries of the 9-j coefficient appearing in quantum theory of angular momentum.     

A note on the generalization of the summation formulas for Appell's and Kampé de Fériet's hypergeometric functions

In the present paper, a summation formula of a general triple hypergeometric series F⁽³⁾(x, y, z) introduced by Srivastava [10] is obtained. A particular case of this formula corresponds to a result of Shah [7] involving Kampé de Fériet's double hypergeometric function which can further be specialized to yield summation formulas of Srivastava [11] and Bhatt [2] for Appell's function F₂.     

An alternative proof of the extended Saalschütz summation theorem for the r + 3Fr + 2(1) series with applications

A simple proof is given of a new summation formula recently added in the literature for a terminating _{r + 3}F_{r + 2}(1) hypergeometric series for the case when r pairs of numeratorial and denominatorial parameters differ by positive integers. This formula represents an extension of the well‐known Saalschütz summation formula for a ₃F₂(1) series. Two applications of this extended summation formula are discussed. The first application extends two identities given by Ramanujan and the second, which also employs a similar extension of the Vandermonde–Chu summation theorem for the ₂F₁ series, extends certain reduction formulas for the Kampé de Fériet function of two variables given by Exton and Cvijovi? & Miller. Copyright © 2015 John Wiley & Sons, Ltd.     

Certain summation and transformation formulas for generalized hypergeometric series

We derive summation formulas for generalized hypergeometric series of unit argument, one of which upon specialization reduces to Minton’s summation theorem. As an application we deduce a reduction formula for a certain Kampé de Fériet function that in turn provides a Kummer-type transformation formula for the generalized hypergeometric function _pF_p(x).     

Telescopic generalizations for two 3 F 2-series identities

By combining a telescopic summation formula with Kummer-Thomae-Whipple transformation, we prove two nonterminating ₃ F ₂(1)-series identities with one of them confirming a conjecture by Milgram (2009) and another one extending a couple of terminating series identities due to Gessel and Stanton (1982).     

Extending the factorization principle to hypergeometric series of general form

It is shown that the formulas of operator factorization of hypergeometric functions obtained in the author’s previous works can be extended to hypergeometric series of the most general form. This generalization does not make the technical apparatus of the factorization method more complicated. As an example illustrating the practical effectiveness of the formulas obtained in the paper, we analyze transformation properties of the Horn seriesG ₃, whose structure is typical for general hypergeometric functions. It is shown that Erdélyi’s transformation formula relating the seriesG ₃ to the Appell functionF ₂, contains erroneous expressions in the arguments ofG ₃. The correct analog of Erdélyi’s formula is found, and some new transformations of the seriesG ₃ are presented. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 573–581, April, 2000.     

On the Mellin transform of products of Bessel and generalized hypergeometric functions

We deduce in an elementary way representations for the Mellin transform of a product of Bessel functions ₀F₁[−a²x²] and generalized hypergeometric functions _pF_p+1[−b²x²] for a,b>0. As a corollary we obtain a transformation formula for _p+1F_p[1] which was discovered by Wimp in 1987 by using Bailey's method for the specialization ₃F₂[1].     

Some Combinatorics of the Hypergeometric Series

Symmetry formulas for the classical hypergeometric series ₂F₁ are proved combinatorially. The idea of the proofs is to find weighted combinatorial structures which form models for each side of the formula and to show how to go from the first to the second model by a ‘weak isomorphism’ (i.e. a sequence of isomorphisms, regroupings and degroupings of structures). This is then applied to the four ₂F₁-families (Meixner, Krawtchouk, Meixner-Pollaczek and Jacobi) of hypergeometric orthogonal polynomials. We give three ‘weakly isomorphic’ models for each family and prove in a completely combinatorial way the 3-terms recurrences for these polynomials.     

On certain Schlömilch-type series

By using a form of the Poisson summation formula together with a generalization due to Srivastava and Exton of the discontinuous integral of Weber and Schafheitlin and the (heretofore not readily available) cosine transform of the hypergeometric function ₁F₂[−b²t²] (b > 0), several new Schlömilch and Fourier-type series are evaluated. By specialization of the latter series numerous results appearing in the literature are obtained in a unified way.     

Reduction and transformation formulas for the Appell and related functions in two variables

In many seemingly diverse areas of applications, reduction, summation, and transformation formulas for various families of hypergeometric functions in one, two, and more variables are potentially useful, especially in situations when these hypergeometric functions are involved in solutions of mathematical, physical, and engineering problems that can be modeled by (for example) ordinary and partial differential equations. The main object of this article is to investigate a number of reductions and transformations for the Appell functions F₁,F₂,F₃, and F₄ in two variables and the corresponding (substantially more general) double‐series identities. In particular, we observe that a certain reduction formula for the Appell function F₃ derived recently by Prajapati et al., together with other related results, were obtained more than four decades earlier by Srivastava. We give a new simple derivation of the previously mentioned Srivastava's formula 12 . We also present a brief account of several other related results that are closely associated with the Appell and other higher‐order hypergeometric functions in two variables. Copyright © 2017 John Wiley & Sons, Ltd.     

Euler transformation formula for multiple basic hypergeometric series of type A and some applications

A multiple generalization of the Euler transformation formula for basic hypergeometric series ₂φ₁ is derived. It is obtained from the symmetry of the reproducing kernel for Macdonald polynomials by a method of multiple principal specialization. As applications, elementary proofs of the Pfaff-Saalschutz summation formula and the Gauss summation formula for basic hypergeometric series in U(n+1) due to S.C. Milne are given. Some other multiple transformation and summation formulas for very-well-poised ₁₀φ₉ and ₈φ₇ series, balanced ₄φ₃ series and ₃φ₂ series are also given.     

Analytic continuation of the Appell function <Emphasis Type="Italic">F</Emphasis><Subscript>1</Subscript> and integration of the associated system of equations in the logarithmic case

The Appell function F ₁ (i.e., a generalized hypergeometric function of two complex variables) and a corresponding system of partial differential equations are considered in the logarithmic case when the parameters of F ₁ are related in a special way. Formulas for the analytic continuation of F ₁ beyond the unit bicircle are constructed in which F ₁ is determined by a double hypergeometric series. For the indicated system of equations, a collection of canonical solutions are presented that are two-dimensional analogues of Kummer solutions well known in the theory of the classical Gauss hypergeometric equation. In the logarithmic case, the canonical solutions are written as generalized hypergeometric series of new form. The continuation formulas are derived using representations of F ₁ in the form of Barnes contour integrals. The resulting formulas make it possible to efficiently calculate the Appell function in the entire range of its variables. The results of this work find a number of applications, including the problem of parameters of the Schwarz–Christoffel integral.     

Gamma series associated to elements satisfying regularized double shuffle relations

Let K be a field of characteristic 0, and R be a commutative K-algebra. Let Φ(x₀,x₁) be an element in R《x₀,x₁》 with regularized double shuffle relations. We define a gamma series ΓΦ(s)∈1+s²R?s? associated to Φ. We prove that the associated beta series is just the image of ΦY(x₀,x₁) in the commutative formal power series ring R?x₀,x₁?, where if Φ=1+Φ₀x₀+Φ₁x₁, then ΦY=1+Φ₁x₁. We also give some equivalent conditions for the reflection formula of the gamma series ΓΦ(s).     

An elliptic hypergeometric integral with <Emphasis Type="Italic">W</Emphasis>(<Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>) symmetry

In this article we give a new transformation between elliptic hypergeometric beta integrals, which gives rise to a Weyl group symmetry of type F ₄. The transformation is a generalization of a series transformation discovered by Langer, Schlosser, and Warnaar (SIGMA 5:055, 2009). Moreover we consider various limits of this transformation to basic hypergeometric functions obtained by letting p tend to 0.     

A new two-term relation for the 3F2 hypergeometric function of unit argument

The elementary manipulation of series together with summations of Gauss, Saalschutz and Dixon are employed to deduce a two-term relation for the hypergeometric function ₃F₂(1) and a summation formula for the same function, neither of which has previously appeared in the literature. The two-term relation has implications in the calculus of finite differences.     

A class of generalized hypergeometric summations

Summations over the positive integers n of the generalized hypergeometric expressions (±1)ⁿ_pF_p+1[−n²x²] (x > 0) are derived in closed form. The specialization p = 0, for example, reduces to known results for Schlömilch series. In addition, we record the apparently not readily available sine and cosine transforms of _pF_p+1[−b²x²] (b > 0), the latter of which is used together with a form of the Poisson summation formula to deduce the aforementioned results.     

Linearization of the product of symmetric orthogonal polynomials

A new constructive approach is given to the linearization formulas of symmetric orthogonal polynomials. We use the monic three-term recurrence relation of an orthogonal polynomial system to set up a partial difference equation problem for the product of two polynomials and solve it in terms of the initial data. To this end, an auxiliary function of four integer variables is introduced, which may be seen as a discrete analogue of Riemann's function. As an application, we derive the linearization formulas for the associated Hermite polynomials and for their continuousq-analogues. The linearization coefficients are represented here in terms of₃ F ₂ and₃Φ₂ (basic) hypergeometric functions, respectively. We also give some partial results in the case of the associated continuousq-ultraspherical polynomials.     

Euler-type transformations for the generalized hypergeometric function r+2Fr+1(x)

We provide generalizations of two of Euler’s classical transformation formulas for the Gauss hypergeometric function extended to the case of the generalized hypergeometric function _r+2 F _r+1(x) when there are additional numeratorial and denominatorial parameters differing by unity. The method employed to deduce the latter is also implemented to obtain a Kummer-type transformation formula for _r+1 F _r+1 (x) that was recently derived in a different way.     

Bilateral inversions and terminating basic hypergeometric series identities

The q-analogue of Legendre inversions is established and generalized to bilateral sequences. They are employed to investigate the dual relations of three basic formulae due to Jackson and Bailey, on balanced ₃?₂-series, well-poised ₈?₇-series and bilateral ₆ψ₆-series. Several terminating well-poised series identities are consequently derived, including the q-Dixon formulae on terminating ₃ψ₃-series and two terminating well-poised ₅ψ₅-series identities due to [F.H. Jackson, Certain q-identities, Quart. J. Math. (Oxford) 12 (1941) 167-172; W.N. Bailey, On the analogue of Dixon’s theorem for bilateral basic hypergeometric series, Quart. J. Math. (Oxford) 1 (1950) 318-320].