Appell Sequences in Analytic Number Theory: Shifted Expansions and Dirichlet L-Functions

Source abstract

We place the Appell shift theorem for asymptotic expansions inside a Borel--Laplace framework and apply it to two classical functions of analytic number theory: the Hurwitz zeta function $ζ(m,q)$ and the Lerch transcendent $Φ(z,m,a)$. The resulting shifted expansions have coefficients given by Appell polynomials evaluated at the shift, with remainders controlled uniformly on compact sets. The generating sequences are identified explicitly in terms of the Taylor coefficients of the Borel kernels. In a second direction, we extend the Appell--Mellin framework of Navas, Ruiz and Varona to Dirichlet $L$-functions. For a primitive non-principal character $χ$ modulo $q$, we construct an explicit Appell--Mellin generating function $A_χ(t)$ whose associated Appell sequence satisfies $R_n^χ(x) = L(-n,χ,x)$, and we prove the functional equation $A_χ(t) = χ(-1)A_χ(-t)$. We further establish five structural properties of the polynomials $R_n^χ$: identification with generalized Bernoulli polynomials, a difference equation, an explicit formula, a Fourier expansion, and the trivial zeros of $L(s,χ)$. To the best of our knowledge, these results do not appear in the literature.