A Modified-Kernel Degenerate $q$-Sumudu Transform Based on the Kernel $\left(1+\lambda/[u]_q\right)^{-t/\lambda}$ and a Laplace-Like Operational Calculus

We introduce a \emph{modified-kernel degenerate $q$-Sumudu transform} whose defining kernel is $\left(1+\lambda/[u]_q\right)^{-t/\lambda}$ and whose scaling is governed by the $q$-number $[u]_q=(1-q^{u})/(1-q)$. Specifically, \[ \mathcal S^{(III)}_{\lambda,q}\{f(t)\}(u) =\frac{1}{[u]_q}\int_{0}^{\infty}\left(1+\frac{\lambda}{[u]_q}\right)^{-t/\lambda} f(t)\,dt, \] whenever the integral converges. By introducing the auxiliary variable $\sigma(u)=\frac{1}{\lambda}\log\!\left(1+\lambda/[u]_q\right)$, we show that the degenerate $q$-Sumudu transform transform is equivalent to the classical Laplace transform in the $\sigma$--domain, which yields a Laplace-like operational calculus in the transform variable $u$. We derive differentiation, convolution and shift theorems, and obtain an explicit inversion formula via the Bromwich integral. The limiting regimes $q\to1^{-}$ and $\lambda\to0^{+}$ recover the corresponding degenerate and $q$-exponential Sumudu/Laplace kernels. As an application, we develop an algebraic solution scheme for constant-coefficient initial value problems and illustrate it with worked examples.