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

Abstract

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.

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

“…Taking p = 2 and substituting formulas ( 8) and ( 13) into above expressions, expansion (30) recovers [4, Theorem 1] or a slightly different version of it if we use (17) instead of (13). If, on the other hand, we take p = 3 and employ ( 13) or (17) in the series for h p (k; a; b) and ( 14) or (20) in the formula for f p (k; a; b) we arrive at several expansions for 4 F 3 (z) which may be new. Similarly, for p = 4 we arrive at expansions for 5 F 4 (z) in terms of Srivastava's F (3) once we employ (15) or (21).…”

“…The computational complexity of this formula does not increase with growing p unlike ( 12) and (19). Particular cases obtained by expressing g p n using ( 13), ( 17), ( 14), (20), (15) or (21) 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.…”

“…Similarly, for p = 4 we arrive at expansions for 5 F 4 (z) in terms of Srivastava's F (3) once we employ (15) or (21). Generalizations of these ideas to the Fox-Wright function were explored in [20].…”

Analytic Normalized Solutions of 2D Fractional Saint-Venant Equations of a Complex Variable

“…Moreover, it converges for all finite values z to the entire function provided ⊳< − 1: In addition, at the boundary |z | = 1, it has the convergence value (see [23])…”

“…Wright's original attentiveness in this function was connected to the asymptotic theory of partitions [24]. The formula ⊲ is generated in [23] by adding a positive parameter θ > 0 as follows:…”