Representations of hypergeometric functions for arbitrary parameter values and their use

Integral Transforms and Special Functions

2020

Self Cite

We derive a convergent expansion of the generalized hypergeometric function p−1 F p in terms of the Bessel functions 0 F 1 that holds uniformly with respect to the argument in any horizontal strip of the complex plane. We further obtain a convergent expansion of the generalized hypergeometric function p F p in terms of the confluent hypergeometric functions 1 F 1 that holds uniformly in any right half-plane. For both functions, we make a further step and give convergent expansions in terms of trigonometric, exponential and rational functions that hold uniformly in the same domains. For all four expansions we present explicit error bounds. The accuracy of the approximations is illustrated with some numerical experiments. 2010

“…for some η > 0 and 0 < φ < π/2. Further, from the asymptotic properties of G p,0 p,p (z) in the neighborhood of z = 0 given in [11,Property 5] we conclude that…”

Section: Preliminaries On the Meijer-nørlund Function And Nørlund's Cmentioning

confidence: 74%

Section: Preliminaries On the Meijer-nørlund Function And Nørlund's Cmentioning

confidence: 99%

Uniformly convergent expansions for the generalized hypergeometric functions _p_–1F_p and _pF_p

Garcia

Martínez

Integral Transforms and Special Functions

2020

Self Cite

“…The purpose of this section is to present two transformations valid when c = b + p with arbitrary p ∈ N. Hence, they cover both degenerate and non-degenerate cases. Some of the coefficients appearing in these transformations can be expressed in terms of Nørlund's coefficients g n (a; b) which were introduced by Nørlund in [23, (1.33)] and investigated in our papers [10, section 2.2], [8,Property 6] and [13, section 2]. For completeness we also give a short and slightly different account here.…”

Section: Miller-paris Transformations: Degenerate Casementioning

confidence: 99%

“…Substituting this expression into (5) and (8) and canceling constant factors we arrive at (37) and (38), respectively. Taking r = 1, m = 2 in Theorem 5 after some elementary computations we arrive at Corollary 4 Suppose e − d − 1 = 0.…”

Section: Theoremmentioning

confidence: 99%

Alternative Approach to Miller-Paris Transformations and Their Extensions

Prilepkina

2020

Trends in Mathematics

Miller-Paris transformations are extensions of Euler's transformations for the Gauss hypergeometric functions to generalized hypergeometric functions of higher-order having integral parameter differences (IPD). In our recent work we computed the degenerate versions of these transformations corresponding to the case when one parameter difference is equal to a negative integer. The purpose of this paper is to present an independent new derivation of both the general and the degenerate forms of Miller-Paris transformations. In doing so we employ the generalized Stieltjes transform representation of the generalized hypergeometric functions and some partial fraction expansions. Our approach leads to different forms of the characteristic polynomials, one of them appears noticeably simpler than the original form due to Miller and Paris. We further present two extensions of the degenerate transformations to the generalized hypergeometric functions with additional free parameters and additional parameters with negative integral differences.

“…where we applied (20) and the last inequality holds as α > 0 by hypothesis. Here M z is the distance between 1 and the straight line segment [0, z].…”

Section: The Fox-wright Function Near Singularitymentioning

confidence: 99%

The Fox-Wright function near the singularity and the branch cut

Journal of Mathematical Analysis and Applications

Prilepkina

2020

Self Cite

The Fox-Wright function is a further extension of the generalized hypergeometric function obtained by introducing arbitrary positive scaling factors into the arguments of the gamma functions in the summand. Its importance comes mostly from its role in fractional calculus although other interesting applications also exist. If the sums of the scaling factors in the top and bottom parameters are equal, the series defining the Fox-Wright function has a finite non-zero radius of convergence. It was demonstrated by Braaksma in 1964 that the Fox-Wright function can then be extended to a holomorphic function in the complex plane cut along a ray from the positive point on the boundary of the disk of convergence to the point at infinity. In this paper we study the behavior of the Fox-Wright function in the neighborhood of this positive singular point. Under certain restrictions we give a convergent expansion with recursively computed coefficients completely characterizing this behavior. We further compute the jump and the average value of the Fox-Wright function on the banks of the branch cut.