A degenerate--\(q\) hybrid Heine equation and Beta--Kernel integral representations

We introduce a degenerate–\(q\) hybrid coefficient series obtained by multiplying a normalised basic-hypergeometric coefficient array by the degenerate rising factorial defined through the Kim–Kim degenerate Gamma function. The normalisation used here deliberately omits the conventional factor \([(-1)^nq^{n(n-1)/2}]^{1+s-r}\) from the fully standard definition of \({}_r\phi_s\); the precise relationship with the standard convention is stated explicitly. Under explicit nonsingularity assumptions on the \(q\)-denominator parameters and the degenerate Gamma factors, we determine the disc of absolute convergence and give a corrected boundary analysis. The coefficient ratio \((\alpha)^*_{\lambda,n+1}/(\alpha)^*_{\lambda,n}\) is encoded as a rational Euler-transport operator, which leads coefficientwise to a hybrid Heine-type \(q\)-difference equation, numerator and denominator contiguity relations, and Jackson \(q\)-derivative identities. In the scalar case we construct two Frobenius-type local series and formulate the connection coefficients only within the corresponding nonresonant Frobenius span. For terminating series we prove genuine degenerate-Gamma moment representations; for nonterminating series the corresponding moment formalism is explicitly interpreted as a meromorphic regularisation rather than as a positive-measure integral. We also state a symmetric Beta-kernel identity with separate complex-analytic and real-positivity hypotheses and record carefully scaled limit transitions.