A Modified-Degenerate Operational Reformulation of Wα,β,ν-Type Exponential, Trigonometric, and Hyperbolic Functions and Their Laplace Transforms

We study a modified-degenerate version of the Wα,β,ν-factorial and the associated exponential, trigonometric, and hyperbolic families obtained by replacing the Euler gamma function with the modified-degenerate gamma function Γλ*, where λ∈(0,1). A main conclusion of this paper is that this construction does not generate a genuinely new transcendental family. Indeed, since Γλ*(s)=bλsΓ(s),bλ=λlog(1+λ), all modified-degenerate W-functions reduce to exact rescalings of their non-degenerate counterparts. The novelty of the present work is therefore operational rather than structural. We formulate this transport principle explicitly, derive the corresponding modified-degenerate Laplace-transform identities directly in the spectral variable s, establish the induced convolution rule, and obtain first-order asymptotic expansions as λ→0+. We further show that the associated W-derivative is a formal coefficient-shift operator, and conjugate it to the non-degenerate one under the scaling map. As an application, we present a complete Volterra integral-equation example with polynomial memory, including an explicit resolvent representation for the case m=1, together with convergence and residual-error checks supporting the numerical illustrations.