Contiguous relations for the Fox–Wright function

Wang, Xiaoxia

doi:10.1016/j.jmaa.2019.02.035

Cited by 4 publications

(6 citation statements)

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“…}, \end{equation}where

p\in \mathbb {Z}^+_0, q\in \mathbb {Z}^+

are indices and

a_i,b_j\in \mathbb {C}

A_i,B_j\in \mathbb {R}

are parameters such that the series converges. This is a very general type of function, enough to cover many diverse special functions as particular cases 30 —such as the original Mittag‐Leffler functions, 35 multi‐index Mittag‐Leffler functions, 36 hypergeometric functions, 37 Bessel functions, 37 and so forth—but it can also be studied in full generality in its own right 38,39 . Thus, we can consider the Fox–Wright function as a useful kernel with broad applicability.…”

Section: Introductionmentioning

confidence: 99%

“…This is a very general type of function, enough to cover many diverse special functions as particular cases 30 -such as the original Mittag-Leffler functions, 35 multi-index Mittag-Leffler functions, 36 hypergeometric functions, 37 Bessel functions, 37 and so forth-but it can also be studied in full generality in its own right. 38,39 Thus, we can consider the Fox-Wright function as a useful kernel with broad applicability. The next question is, using this function, how exactly do we wish to define fractional integral and fractional derivative operators?…”

mentioning

confidence: 99%

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A catalogue of semigroup properties for integral operators with Fox–Wright kernel functions

2022

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The Fox-Wright function is a very general form of function, covering many families of special functions as particular cases. Any special function can be used as a kernel for a fractional integral operator, but which of these operators will satisfy desiderata such as a semigroup property for composition? This paper provides a rigorous categorisation of all such fractional integral operators which have a semigroup property in any of their parameters. We discover that nearly all possible semigroup properties arise from the Chu-Vandermonde identity, with the Prabhakar fractional calculus emerging as one special case. For any integral operator with such a semigroup property, it is possible to construct a complete model of fractional calculus, including both integral and derivative operators which interact with each other in a natural way.

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“…}, \end{equation}where

p\in \mathbb {Z}^+_0, q\in \mathbb {Z}^+

are indices and

a_i,b_j\in \mathbb {C}

A_i,B_j\in \mathbb {R}

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

A catalogue of semigroup properties for integral operators with Fox–Wright kernel functions

2022

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“…See also [39, section 6] for an application in information theory. It has been a recent surge of interest in the Fox-Wright function as witnessed by the articles [16,36,37,43,44,45,48,52,54]. The papers [16,54] establish summation formulas for the Fox-Wright function using combinatorial inversion formulas.…”

Section: Introductionmentioning

confidence: 99%

“…The papers [16,54] establish summation formulas for the Fox-Wright function using combinatorial inversion formulas. In [52] the author used a somewhat opposite approach by first developing contiguous relations for the Fox-Wright function and then employing them to prove Hagen-Rothe convolutions from combinatorics. Paris [43,44,45] discusses exponentially small asymptotics for p Ψ q when ∆ > −1.…”

Section: Introductionmentioning

confidence: 99%

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

Karp

Prilepkina

2020

Journal of Mathematical Analysis and Applications

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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.

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“…The usage of the function is seen in different branches of science. Many of its properties are documented for years while the related scientific investigation is still an active area of research (Wang, 2019;Wei, 2019). In the context of the current article, we streamline our focus on…”

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confidence: 99%