Mersenne-Lerch Interpolation Values and Apostol-Mersenne Polynomial Families

This paper introduces Bernoulli-type and Euler-type Mersenne-Lerch interpolation families associated with Apostol-Mersenne polynomial families. Their construction is based on the Mersenne translation polynomials $P_{n,M}(x;m)$, defined by $$e_M^{xt}(e_M^t)^m = \sum_{n=0}^{\infty}P_{n,M}(x;m)\frac{t^n}{M_n!}.$$ Explicit formulas for these polynomials are derived, including an expansion in terms of M-Stirling numbers of the second kind. At nonpositive integers, the resulting interpolation values recover the Apostol-Mersenne-Bernoulli and Apostol-Mersenne-Euler polynomials of order $r$. Derivative, integral, addition, and difference formulas are also obtained for these values. A comparison with $q$-calculus shows that the M-factorial, M-binomial coefficient, M-derivative, and M-exponential are obtained by setting $q=2$ in the corresponding $q$-calculus expressions.