From Umbral Hyperbolic Integrals to a Cotangent Coefficient Formula for the Mittag Leffler Polynomials
Abstract
Let gn(x) be the Mittag--Leffler polynomials defined by ∑n≥0gn(x)tn=1/2((1+t)/(1−t))x. We derive the coefficient formula gn(x)=1/n∑j=0⌊ (n−1)/2⌋ (−1)j(2n−2j−1)/((n−2j−1)!) [u2j](ucot u)n xn−2j, equivalently [un−r](ucot u)n =(−1)(n−r)/2(n(r−1)!)/(2r−1)[xr]gn(x). The identity is suggested by comparing an umbral representation of the hyperbolic tangent integral K(a,b)=∫0∞ (tanha x)/xb 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 Im,nβ=∫0∞(tanhm x)/(xncosh x) dx. Defining qm(x) by ∑m≥0qm(x)tm=(1−t2)−1/2exp(2xartanht), we prove [um−r]u/(sin u)(ucot u)m =(−1)(m−r)/2r!/2r[xr]qm(x). This gives a parallel umbral representation for the beta integrals. The family qm is a shifted c=−1,β=1 specialization of the classical Meixner family, yielding a beta/Meixner counterpart to the zeta/Mittag--Leffler correspondence.
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BibTeX
@misc{waffo2026from,
title = {{From Umbral Hyperbolic Integrals to a Cotangent Coefficient Formula for the Mittag Leffler Polynomials}},
author = {Luc Ramsès Talla Waffo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16092},
eprint = {2609.16092},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16092},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From Umbral Hyperbolic Integrals to a Cotangent Coefficient Formula for the Mittag Leffler Polynomials
AU - Luc Ramsès Talla Waffo
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.16092
UR - https://arxiv.org/abs/2609.16092
AB - Let g_n(x) be the Mittag--Leffler polynomials defined by ∑_n≥0g_n(x)t^n=1/2((1+t)/(1−t))^x. We derive the coefficient formula g_n(x)=1/n∑_j=0^⌊ (n−1)/2⌋ (−1)^j(2^n−2j−1)/((n−2j−1)!) [u^2j](ucot u)^n x^n−2j, equivalently [u^n−r](ucot u)^n =(−1)^(n−r)/2(n(r−1)!)/(2^r−1)[x^r]g_n(x). The identity is suggested by comparing an umbral representation of the hyperbolic tangent integral K(a,b)=∫_0^∞ (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^β=∫_0^∞(tanh^m x)/(x^ncosh x) dx. Defining q_m(x) by ∑_m≥0q_m(x)t^m=(1−t^2)^−1/2exp(2xartanht), we prove [u^m−r]u/(sin u)(ucot u)^m =(−1)^(m−r)/2r!/2^r[x^r]q_m(x). This gives a parallel umbral representation for the beta integrals. The family q_m is a shifted c=−1,β=1 specialization of the classical Meixner family, yielding a beta/Meixner counterpart to the zeta/Mittag--Leffler correspondence.
ER -