Generalized Hypergeometric Functions Research Articles

Finite summation formulas of Srivastava’s general triple hypergeometric function

This paper continues the study of finite summation formulas of multivariable hypergeometric functions. Earlier, Wang and Chen [Finite summation formulas of double hypergeometric functions, Int. Transf. Spec. Funct. 28(3) (2016) 239–253] established finite summation formulas of double hypergeometric functions by using some well-known summation theorems. We present here finite summation formulas for Srivastava’s general triple hypergeometric function. Certain particular cases leading to finite summation formulas for Lauricella functions and Srivastava’s triple hypergeometric functions are also presented.

Accelerating indefinite hypergeometric summation algorithms

Let K be a field of characteristic zero, x an independent variable, E the shift operator with respect to x, i.e., Ef ( x ) = f ( x + 1) for an arbitrary f ( x ). Recall that a nonzero expression F ( x ) is called a hypergeometric term over K if there exists a rational function r ( x ) ∈ K( x ) such that F ( x + 1)/ F ( x ) = r ( x ). Usually r ( x ) is called the rational certificate of F ( x ). The problem of indefinite hypergeometric summation (anti-differencing) is: given a hypergeometric term F ( x ), find a hypergeometric term G ( x ) which satisfies the first order linear difference equation ( E − 1) G ( x ) = F ( x ). (1) If found, write Σ x F ( x ) = G ( x ) + c , where c is an arbitrary constant.

A q-microscope for supercongruences

By examining asymptotic behavior of certain infinite basic (q-) hypergeometric sums at roots of unity (that is, at a ‘q-microscopic’ level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial (super)congruences for truncated ordinary hypergeometric sums, which have been observed numerically and proven rarely. A typical example includes derivation, from a q-analogue of Ramanujan's formula∑n=0∞(4n2n)(2nn)228n32n(8n+1)=23π, of the two supercongruencesS(p−1)≡p(−3p)(modp3)andS(p−12)≡p(−3p)(modp3), valid for all primes p>3, where S(N) denotes the truncation of the infinite sum at the N-th place and (−3⋅) stands for the quadratic character modulo 3.

Gauss's theorem and harmonic number summation formulae with certain mathematical constants

ABSTRACTBy computing the higher-order derivative of the famous Gauss hypergeometric summation theorem, we derive many harmonic number summation formulae with certain mathematical constants such as π, the Euler-Mascheroni constant γ, the Catalan constant G and the Apéry constant .

Bounds of coefficients for classes of analytic functions related to hypergeometric functions

The main purpose of the present paper is to give some sharp coefficients bounds for a certain class of univalent analytic functions in unit open disk, which was defined by using principle of differential subordination and generalized hypergeometric function. As applications, we investigate the almost starlike-type functions, parabolic starlike-type functions and uniformly convex-type functions with conic domain. Our results extend some earlier works related to Ma-Minda starlike and convex functions.

Infinite summation formulas related to Riemann Zeta function from hypergeometric series

Applying Gauss and Watson’s famous hypergeometric summation theorems, the authors establish two pattern infinite summation formulas involving generalized harmonic numbers related to Riemann Zeta function.

Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems

We list $A_n$, $C_n$ and $D_n$ extensions of the elliptic WP Bailey transform and lemma, given for $n=1$ by Andrews and Spiridonov. Our work requires multiple series extensions of Frenkel and Turaev's terminating, balanced and very-well-poised ${}_{10}V_9$ elliptic hypergeometric summation formula due to Rosengren, and Rosengren and Schlosser. In our study, we discover two new $A_n$ ${}_{12}V_{11}$ transformation formulas, that reduce to two new $A_n$ extensions of Bailey's $_{10}\phi_9$ transformation formulas when the nome $p$ is $0$, and two multiple series extensions of Frenkel and Turaev's sum.

Summations of linear recurrent sequences

We give an extension of Sister Celine’s method of proving hypergeometric sum identities that allows it to handle a larger variety of input summands. In particular, we extend the summand to powers of a C-finite sequence times a hypergeometric term. We then apply this to several problems. Some of these applications give new results, and some reprove already known results in an automated way.

Two-sided inequalities for the Struve and Lommel functions

Mathematical inequalities and other results involving such widely- and extensively-studied special functions of mathematical physics and applied mathematics as (for example) the Bessel, Struve and Lommel functions as well as the associated hypergeometric functions are potentially useful in many seemingly diverse areas of applications, especially in situations in which these functions are involved in solutions of mathematical, physical and engineering problems which can be modeled by ordinary and partial differential equations. With this objective in view, our present investigation is motivated by some open problems involving inequalities for a number of particular forms of the hypergeometric function 1F2(a; b, c; z). Here, in this paper, we apply a novel approach to such problems and obtain presumably new two-sided inequalities for the Struve function, the associated Struve function and the modified Struve function by first investigating inequalities for the general hypergeometric function 1F2(a; b, c; z). We also briefly discuss the analogous new inequalities for the Lommel function under some conditions and constraints. Finally, as special cases of our main results, we deduce several inequalities for the modified Lommel function and the normalized Lommel function.

Extensions of the classical theorems for very well-poised hypergeometric functions

The classical summation and transformation theorems for very well-poised hypergeometric functions, namely, $_{5}F_4(1)$ summation, Dougall's $_{7}F_6(1)$ summation, Whipple's $_{7}F_6(1)$ to $_{4}F_3(1)$ transformation and Bailey's $_{9}F_8(1)$ to $_{9}F_8(1)$ transformation are extended. These extensions are derived by applying the well-known Bailey's transform method along with the classical very well-poised summation and transformation theorems for very well-poised hypergeometric functions and the Rakha and Rathie's extension of the Saalsch\"{u}tz's theorem. To show importance and applications of the discovered extensions, a number of special cases are pointed out, which leads not only to the extensions of other classical theorems for very well-poised and well-poised hypergeometric functions but also generate new hypergeometric summations and transformations.

Integrals Involving Aleph Function and Wright’s Generalized Hypergeometric Function

The aim of this paper is to establish certain integrals involving product of the Aleph function with Srivastava’s polynomials and Fox-Wright’s Generalized Hypergeometric function. Being unified and general in nature, these integrals yield a number of known and new results as special cases. For the sake of illustration, four corollaries are also recorded here as special case of our main results.

Generalized harmonic number summation formulae via hypergeometric series and digamma functions

By computing the derivatives of five classical hypergeometric summation theorems, and applying the related properties of the digamma function, we derive a large number of closed summation formulae for generalized harmonic numbers.

Fractional Integral Operators Characterized by Some New Hypergeometric Summation Formulas

Abstract The purpose of this paper is to study generalized fractional integral operators whose kernels involve a very special class of generalized hypergeometric functions r+2 F r+1. The importance of these operators stems from the summability of this special class of generalized hypergeometric functions at unit argument which is realized in the present paper by making use of the Karlsson-Minton type summation theorems. We obtain for these operators several results including the generalized fractional integrals of the H-function, certain mapping properties and their Mellin transforms. Applications to inequalities are also pointed out.

Inner bounds for the extreme zeros of 3F2 hypergeometric polynomials

ABSTRACTZeilberger's celebrated algorithm finds pure recurrence relations (w.r.t. a single variable) for hypergeometric sums automatically. However, in the theory of orthogonal polynomials and special functions, contiguous relations w.r.t. several variables exist in abundance. We modify Zeilberger's algorithm to generate unknown contiguous relations that are necessary to obtain inner bounds for the extreme zeros of orthogonal polynomial sequences with hypergeometric representations. Using this method, we improve previously obtained upper bounds for the smallest and lower bounds for the largest zeros of the Hahn polynomials and we identify inner bounds for the extreme zeros of the Continuous Hahn and Continuous Dual Hahn polynomials. Numerical examples are provided to illustrate the quality of the new bounds. Without the use of computer algebra such results are not accessible. We expect our algorithm to be useful to compute useful and new contiguous relations for other hypergeometric functions.

Recursion formulas for the Srivastava–Daoust and related multivariable hypergeometric functions

This paper concludes the study of recursion formulas of multivariable hypergeometric functions. Earlier in [V. Sahai and A. Verma, Recursion formulas for multivariable hypergeometric functions, Asian–Eur. J. Math. 8 (2015) 50, 1550082], the authors have given the recursion formulas for three variable Lauricella functions, Srivastava’s triple hypergeometric functions and [Formula: see text]-variable Lauricella functions. Further, in [V. Sahai and A. Verma, Recursion formulas for Recursion formulas for Srivastava’s general triple hypergeometric functions, Asian–Eur. J. Math. 9 (2016) 17, 1650063], we have obtained recursion formulas for Srivastava general triple hypergeometric function [Formula: see text]. We present here the recursion formulas for generalized Kampé de Fériet series and Srivastava and Daoust multivariable hypergeometric function. Certain particular cases leading to recursion formulas of certain generalized hypergeometric function of one variable, certain Horn series, Humbert’s confluent hypergeometric series and some confluent forms of Lauricella series in [Formula: see text]-variables are also presented.

Basic hypergeometry of supersymmetric dualities

We introduce several new identities combining basic hypergeometric sums and integrals. Such identities appear in the context of superconformal index computations for three-dimensional supersymmetric dual theories. We give both analytic proofs and physical interpretations of the presented identities.

Recursion formulas for Srivastava’s general triple hypergeometric function

This paper continues the study of recursion formulas of multivariable hypergeometric functions. In [Recursion formulas for multivariable hypergeometric functions, Asian-Eur. J. Math. 8(4) (2015) Article ID: 1550082, 50 pp.], the authors have given the recursion formulas for three-variable Lauricella functions, three Srivastava’s triple hypergeometric functions and four [Formula: see text]-variable Lauricella functions. We present here the recursion formulas for the general triple hypergeometric function.

Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation

We use elliptic Taylor series expansions and interpolation to deduce a number of summations for elliptic hypergeometric series. We extend to the well-poised elliptic case results that in the $q$-case have previously been obtained by Cooper and by Ismail and Stanton. We also provide identities involving S. Bhargava's cubic theta functions.

New Prospective of Hypergeometric Summation

New Prospective of Hypergeometric Summation

Formal Proofs of Hypergeometric Sums

Algorithmic methods can successfully automate the proof, and even the discovery, of a large class of identities involving sums of hypergeometric terms. In particular, the Wilf-Zeilberger (WZ) algorithm is a uniform framework for a substantial class of hypergeometric summation problems. This algorithm can produce a rational function certificate that can, on the face of it, be used to verify the result by routine algebraic manipulations, independently of the working of the algorithm that discovered it. It is therefore very natural to consider using this certificate to produce, by automated means, a rigorous deductive proof in an interactive theorem prover. However, naive presentations of the WZ method tend to gloss over trivial-looking but rather knotty questions about zero denominators, which makes their rigorous formalization tricky and their ultimate logical justification somewhat obscure. We describe how we have handled these difficulties to produce rigorous WZ proofs inside the HOL Light theorem prover.