K-Humbert confluent hypergeometric functions \(\Phi_{1,k}\), \(\Phi_{2,k}\), and \(\Phi_{3,k}\)

Volume 35, Issue 3, pp 348--361 https://dx.doi.org/10.22436/jmcs.035.03.07

Publication Date: May 29, 2024 Submission Date: March 23, 2024 Revision Date: April 10, 2024 Accteptance Date: April 29, 2024

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Authors

M. Arshad - Department of Mathematics and Statistics, University of Agriculture, Faisalabad 38000, Pakistan. M. Usman - Department of Mathematics and Statistics, University of Agriculture, Faisalabad 38000, Pakistan. M. Z. Iqbal - Department of Mathematics and Statistics, University of Agriculture, Faisalabad 38000, Pakistan. A. Ali - Department of Mathematics, Government Graduate College of Science, Faisalabad, Pakistan.

Abstract

The main aim of this work is to introduce the k-Humbert confluent hypergeometric functions \(\Phi_{1,k}\), \(\Phi_{2,k}\), \(\Phi_{3,k}\) and developed some formulae by using the idea of k-calculus. k-Humbert confluent hypergeometric functions are an extension of the classical confluent hypergeometric functions, incorporating an additional parameter, k, which enriches their analytical properties and applicability in diverse mathematical contexts. We introduce new results by applying $q$-derivative operator on k-Humbert confluent hypergeometric functions \(\Phi_{1,k}\), \(\Phi_{2,k}\), and \(\Phi_{3,k}\). Contiguous function relations of various types, (q, k)-recurrence relations, and $q$-derivatives formulae are constructed.

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Keywords

• Humbert confluent hypergeometric functions
• $q$-derivative
• (q, k)-recurrence relations and contiguous function

MSC

•  33C15
•  33C20

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