From Umbral Hyperbolic Integrals to a Cotangent Coefficient Formula for the Mittag Leffler Polynomials

Let $g_n(x)$ be the Mittag–Leffler polynomials defined by $$\sum _{n\ge 0}{g}_n(x)t^n=\frac{1}{2}{\left(\frac{1+t}{1-t}\right)}^x\mathrm{.}$$ We derive the coefficient formula $$g_n(x)=\frac{1}{n}\sum _{j=0}^{\lfloor (n-1)\mathrm{/}2\rfloor }(-1{)}^j\frac{2^{n-2j-1}}{(n-2j-1)!}[u^{2j}](u\cot u{)}^n{x}^{n-2j},$$ equivalently $$[u^{n-r}](u\cot u{)}^n=(-1{)}^{(n-r)\mathrm{/}2}\frac{n(r-1)!}{2^{r-1}}[x^r]g_n(x)\mathrm{.}$$ The identity is suggested by comparing an umbral representation of the hyperbolic tangent integral $$K(a,b)={\int }_0^{\mathrm{\infty }}\frac{{\tanh }^{a}x}{x^b}dx$$ with a compact cotangent-coefficient formula, but is proved independently from the generating function by formal Lagrange–Bürmann inversion. Substitution of the proved bridge back into the integral formula then yields a rigorous derivation of the umbral Mittag–Leffler representation. We also develop a Dirichlet–beta analogue for $$I_{m,n}^{\beta }={\int }_0^{\mathrm{\infty }}\frac{{\tanh }^{m}x}{x^n\cosh x}dx\mathrm{.}$$ Defining $q_m(x)$ by $$\sum _{m\ge 0}{q}_m(x)t^m=(1-t^2{)}^{-1\mathrm{/}2}\exp (2x\mathrm{artanh}t),$$ we prove $$[u^{m-r}]\frac{u}{\sin u}(u\cot u{)}^m=(-1{)}^{(m-r)\mathrm{/}2}\frac{r!}{2^r}[x^r]q_m(x)\mathrm{.}$$ This gives a parallel umbral representation for the beta integrals. The family $q_m$ is a shifted $c=-1,\beta =1$ specialization of the classical Meixner family, yielding a beta/Meixner counterpart to the zeta/Mittag–Leffler correspondence.