**Title:** Decomposition Formulas for Second-Order Quadruple Gaussian Hypergeometric Series by Means of Operators *H*(α,β) and *H*(α,β)

**Montes Taurus J. Pure Appl. Math.** / ISSN: 2687-4814

**Article ID:** MTJPAM-D-21-00019; **Volume 4 / Issue 3 / Year 2022 (Special Issue)**, Pages 41-60

**Document Type:** Research Paper

**Author(s):** Anvar Hasanov ^{a} , Ainur Ryskan ^{b} , Junesang Choi ^{c}

^{a}Institute of Mathematics, Uzbek Academy of Sciences, 81 Mirzo-Ulugbek street, Tashkent 700170, Uzbekistan

^{b}National Pedagogical University, 86 Tole bi street, Almaty 0500012, Kazakhstan

^{c}Department of Mathematics, Dongguk University, Gyeongju 38066, Republic of Korea

Received: 2 February 2021, Accepted: 15 July 2021, Published: 30 July 2021.

**Corresponding Author:** Junesang Choi (Email address: junesang@dongguk.ac.kr)

**Full Text:** PDF

**Abstract**

Numerous decomposition formulas for various hypergeometric functions of several variables have been offered. In this paper, we aim to establish
symbolic operator identities and decomposition formulas for second-order quadruple Gaussian hypergeometric series associated with Appell functions and Saran hypergeometric functions
by mainly using mutually inverse symbolic operators *H*(α,β) and *H*(α,β), which were introduced in an earlier work. Mellin-Barnes type contour integrals are employed for proofs of the operator identities. Also we determine the regions of convergence of the 14 quadruple Gaussian hypergeometric series.

**Keywords:** Hypergeometric functions, Multiple hypergeometric functions, Inverse pairs of symbolic operators, Decomposition formulas, Mellin-Barnes contour integrals

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