Abstract
In this article, we introduce some of the mathematical properties of the second Appell hypergeometric matrix function F2(A, B1, B2, C1, C2; z, w) including integral representations, transformation formulas, and series formulas.
Highlights
Appell defined and studied in [1,2,3] four kinds of double series of two variables z, w as generalizations of the hypergeometric series: F(α, β, c; z) ∞ (α)n(β)nzn, n 0 (c)n(1)n (1)where z is a main variable in the unit disk {z ∈ C: |z| < 1}, α, β, c are complex parameters with c ≠ 0, − 1, − 2, − 3, . . ., and (α)n α(α + 1)(α + 2). . . (α + n − 1) (n ∈ N) and (α)0 = 1
The Appell hypergeometric series F2 arises frequently in various physical and chemical applications ([8,9,10,11]). e exact solutions of number of problems in quantum mechanics have been given [6, 7, 9, 12] in terms of Appell’s function F2. They can find some results of the classical second Appell hypergeometric function F2 in [13,14,15,16,17]
The extension of the classical Appell hypergeometric functions Fs, s {1, 2, 3, 4}, to the Appell hypergeometric matrix functions has been a subject of intensive studies [26,27,28,29,30]. e purpose of the present work is to study the second Appell hypergeometric matrix function F2(A, B1, B2, C1, C2; z, w) on the domain (z, w) ∈ C2: |z| + |w| < 1, with square matrix valued parameters A, B1, B2, C1, and C2 in Cd×d
Summary
They can find some results of the classical second Appell hypergeometric function F2 in [13,14,15,16,17]. E purpose of the present work is to study the second Appell hypergeometric matrix function F2(A, B1, B2, C1, C2; z, w) on the domain (z, w) ∈ C2: |z| + |w| < 1, with square matrix valued parameters A, B1, B2, C1, and C2 in Cd×d. If E and F are positive stable matrices in Cd×d and EF FE, the Beta matrix function is well defined by
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