Functional Quantum Calculus: Theory and Applications via q(t) and h(t)Operators
DOI:
https://doi.org/10.48165/jmmfc.2026.3103Keywords:
q – calculus, Differential equations, Geometric function theory, Special functions.Abstract
This paper introduces a novel framework for functional quantum calculus based on generalized q(t) and h(t) calculi, with applications to geometric function theory and special functions. We define q(t) – exponential, Gamma, Beta functions, hypergeometric series, Hermite polynomials, and starlike/convex functions, along with formulating q(t)- and h(t)- differential equations. Key contributions include establishing the relation between h(t)- and q(t) – derivatives and the fundamental theorem of h(t) – calculus. We also prove orthogonality for q(τ) – Hermite polynomials and provide numerical examples.
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