K-Humbert confluent hypergeometric functions Φ 1 , k , Φ 2 , k , and Φ 3 , k

The main aim of this work is to introduce the k-Humbert confluent hypergeometric functions Φ 1,k , Φ 2,k , Φ 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 Φ 1,k , Φ 2,k , and Φ 3,k .Contiguous function relations of various types, (q, k)-recurrence relations, and q-derivatives formulae are constructed.