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We solve the second-order linear differential equation called thek-hypergeometric differential equation by using Frobenius method around all its regular singularities. At each singularity, we find 8 solutions corresponding to the different cases for parameters and modified our solutions accordingly.

In the paper, the authors present some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function via some classical inequalities such as Chebychev’s inequality for synchronous (or asynchronous, respectively) mappings, give a new proof of the log-convexity of the extended gamma function by using the Hölder inequality, and introduce a Turán type mean inequality for the Kummer confluent k-hypergeometric function.

In the research paper, the authors exploit the definition of a new class of fractional integral operators, recently proposed by Jarad et al. (Adv. Differ. Equ. 2017:247, 2017), to define a new class of generalized k-fractional integral operators and develop a generalization of the reverse Minkowski inequality involving the newly introduced fractional integral operators. The two new theorems correlating with this inequality, including statements and verifications of other inequalities via the suggested k-fractional conformable integral operators, are presented.

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The main objective of this paper is to derive contiguous function relations or recurrence relations and obtainan integral representation Appell -series , where .Keywords: Pochhammer -symbol, -gamma function, -beta function, Contiguous functions, Appell -series. Contribution/ OriginalityThis study originates a new formula for Appell's series in the form of a new symbol and contributes for deriving contiguous function relations, obtaining an integral representation of the Appell's series in terms of said symbol .

In this research work, our aim is to determine the contiguous function relations for -hypergeometric functions with one parameter corresponding to Gauss fifteen contiguous function relations for hypergeometric functions and also obtain contiguous function relations for two parameters. Throughout in this research paper, we find out the contiguous function relations for both the cases in terms of a new parameter > 0. Obviously if → 1, then the contiguous function relations for -hypergeometric functions are Gauss contiguous function relations.

In this paper, we introduce the (k, s)-fractional integral and differential operators involving k-Mittag-Leffler function E δ k,ρ,β (z) as its kernel. Also, we establish various properties of these operators. Further, we consider a number of certain consequences of the main results.