A Study of Extended Beta, Gauss and Confluent Hypergeometric Functions

A novel Kind of the multi-index Beta, Gauss, and confluent hypergeometric functions

“…, then we get the extended beta function defined by Ghayasuddin et al [7], which again for s = 2, by setting a 1 = λ, a 2 = 0, and b 1 = b 2 = 1, yields the extended beta function introduced by Shadab et al [19].…”

“…More recently, Ghayasuddin et al [7] introduced the following multi-index generalization of the beta function by making use of the multi-index Mittag-Leffler function:…”

“…Remark 6.1. On setting c 1 = c 2 = • • • = c s = 1, (6.1) and (6.2) reduce to the known extensions of Gauss and confluent hypergeometric functions defined by Ghayasuddin et al[7], which further for s = 2, with a 1 = 1, a 2 = 0, and b 1 = b 2 = 1, yields the known extensions of Gauss and confluent hypergeometric functions defined by Shadab et al[19]. The following integral representations of our multi-index Gauss and confluent hypergeometric functions holds true:F (a 1 ,...,a s ,b 1 ,...,b s ) (c 1 ,...,c s ),τ (κ 1 , κ 2 ; κ 3 ; u)…”

A Novel Kind of the Multi-Index Whittaker Function and Its Certain Properties

“…A list of Whittaker functions have been explored (see, for example, [1], [3], [6], [7], [18], [26], [27] and the references therein). In this sequel, we expect to present a new generalization of the Whittaker function of the first kind as far as summed up confluent hypergeometric function introduced by Ghayasuddin et al [10].…”

“…More recently, Ghayasuddin et al [10] presented a further extension of the Beta function by involving of the multi-index (2s-parameter) Mittag-Leffler function:…”