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In this article, we consider a unified generalized version of extended Euler's Beta function's integral form a involving Macdonald function in the kernel. Moreover, we establish functional upper and lower bounds for this extended Beta function. Here, we consider the most general case of the four-parameter Macdonald function K ν + 1 2 p t − λ + q (1 − t) − μ when λ ≠ μ in the argument of the kernel. We prove related bounding inequalities, simultaneously complementing the recent results by Parmar and Pogány in which the extended Beta function case λ = μ is resolved. The main mathematical tools are integral representations and fixed-point iterations that are used for obtaining the stationary points of the argument of the Macdonald kernel function K ν + 1 2 . [ABSTRACT FROM AUTHOR]

The principal aim of this paper is to introduce the extended Voigt-type function V μ , ν (x , y) and its counterpart extension W μ , ν (x , y) , involving the Neumann function Y ν in the kernel of the representing integral. The newly defined integral reduces to the classical Voigt functions K (x , y) and L (x , y) , and to their generalizations by Srivastava and Miller, by the unification of Klusch. Following an approach by Srivastava and Pogány, we also present the multiparameter and multivariable versions V μ , ν (r) (x , y) , W μ , ν (r) (x , y) and the r positive integer of the initial extensions V μ , ν (x , y) , W μ , ν (x , y) . Several computable series expansions are obtained for the discussed Voigt-type functions in terms of Humbert confluent hypergeometric functions Ψ 2 (r) . Furthermore, by transforming the input extended Voigt-type functions by the Grünwald–Letnikov fractional derivative, we establish representation formulae in terms of the associated Legendre functions of the second kind Q η − ν in the two-parameter and two-variable cases. Finally, functional bounding inequalities are given for V μ , ν (x , y) and W μ , ν (x , y) . Particularly interesting results are presented for the Neumann function Y ν and for the Struve H ν function in the form of several functional bounds. The article ends with a thorough discussion and closing remarks. [ABSTRACT FROM AUTHOR]

Recently the authors obtained several Laplace transforms of convolution type integrals involving Kummer’s function 1F1 [Appl. Anal. Discrete Math., 2018, 12(1), 257-272]. In this paper, the authors aim at presenting several new and interesting Laplace transforms of convolution type integrals involving product of two special generalized hypergeometric functions pFp by employing classical summation theorems for the series 2F1, 3F2, 4F3 and 5F4 available in the literature.

Our main goal in this article is to present both upper and lower bounds for the (p,v) - extended Gauss' hypergeometric function and the related confluent hypergeometric (or Kummer's) function, the modified Bessel function of the second kind, with the extended Gautschi-Pinelis inequality (upper bounds) and with the aid of the classical Bernoulli inequality (lower bounds) and also inferring associated functional bounds for the (p,v)-extended Beta function. [ABSTRACT FROM AUTHOR]

Our purpose in this paper is to consider a generalized form of the extended Hurwitz-Lerch Zeta function. For this extended Hurwitz-Lerch Zeta function, we obtain some classical properties which includes various integral representations, a differential formula, Mellin transforms and certain generating relations. We further consider an application to probability distributions and also point out some important special cases of the main results.

Integral form expressions are obtained for the Mathieu-type series and for their associated alternating versions, the terms of which contain a (p , ν) -extended Gauss hypergeometric function. Contiguous recurrence relations are found for the Mathieu-type series with respect to two parameters, and finally, particular cases and related bounding inequalities are established. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce general families of the extended Mathieu‐type power series and present a number of potentially useful integral representations of several general families of the extended Mathieu‐type power series in a unified manner. Relationships of the extended Mathieu‐type functional power series with the generalized Hurwitz–Lerch zeta function is also considered. Various other properties, mainly, Mellin transform and Hankel transform, and fractional derivative formulae are derived for the extended Mathieu series. A pair of the bounding inequalities are established for the extended Mathieu‐type series. As an application of newly defined function, we present a systematic study of probability density function and distribution function associated with the general extended Mathieu‐type power series. In particular, the mathematical expectation and variance of the distribution are derived. Finally, we prove some properties of monotonicity, convexity, and Turán‐type inequalities for the general extended Mathieu‐type power series. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to present closed integral form expressions for the Mathieu-type a-series and for the associated alternating versions whose terms contain a generalized p-extended Gauss' hypergeometric function. Related bounding inequalities for the p-generalized Mathieu-type series are also obtained. Finally, a set of various (known or new) special cases and consequences of the results earned are presented. [ABSTRACT FROM AUTHOR]

Motivated by recent generalizations of classical theorems for the series 2F1 [Integral Transform. Spec. Funct. 229(11), (2011), 823-840] and interesting Laplace transforms of Kummer's confluent hypergeometric functions obtained by Kim et al. [Math. Comput. Modelling 55 (2012), 1068-1071], first we express generalized summations theorems in explicit forms and then by employing these, we derive various new and useful Laplace transforms of convolution type integrals by using product theorem of the Laplace transforms for a pair of Kummer's confluent hypergeometric function. [ABSTRACT FROM AUTHOR]

Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [30] and the second Appell function [6], we introduce here the incomplete Lauricella functions γ(n)A and Γ(n)A of n variables. We then systematically investigate several properties of each of these incomplete Lauricella functions including, for example, their various integral representations, finite summation formulas, transformation and derivative formulas, and so on. We provide relevant connections of some of the special cases of the main results presented here with known identities. Several potential areas of application of the incomplete hypergeometric functions in one and more variables are also pointed out. [ABSTRACT FROM AUTHOR]