From Inverse Vandermonde Matrices to the Shift Structure of the Dirichlet Eta and Beta Functions
Abstract
We investigate the asymptotic behavior of the inverse of Vandermonde matrices with nodes drawn from the quadratic sequences u n = n 2 and u n = (n -1 2 ) 2 . We derive new limiting formulas for the entries of the inverses, expressed through the even and odd central factorial numbers. These formulas are then used to study the asymptotic behavior of a ratio of determinants involving a modified Vandermonde matrix. For each node sequence, the corresponding determinant ratio converges uniformly on compact subsets of the complex plane to an explicit expression involving the Dirichlet eta or beta function.
We then study a family of related functions and, using an inversion theorem, exhibit their shift structure: the shifted values of the eta and beta functions are recovered as finite linear combinations of the family, which yields an integral form of these values as integrals of polynomials related to the Bernoulli and Euler polynomials against generalized (half-)Clausen functions. A unified form of the shift representation, valid for an arbitrary base point z with Re(z) > 0, brings the eta and beta families under a single theorem. On the critical line the representation is unitary and the values on the two sides are complex conjugates.
Related
- Multi-Index Recovery for Total-Degree Multivariate Vandermonde Matrices
- Spectra of Multivariate Vandermonde Matrices and Min-max Error Bounds for the High-dimensional Super-resolution Problem
- The First Moment of Dirichlet L-functions over Generators of the Character Group
- Large values of cubic and quartic Dirichlet L-functions of prime moduli
- Central non-vanishing of Dirichlet L-functions
BibTeX
@misc{iovleff2026from,
title = {{From Inverse Vandermonde Matrices to the Shift Structure of the Dirichlet Eta and Beta Functions}},
author = {Serge Iovleff},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26408},
eprint = {2610.26408},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05804361v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From Inverse Vandermonde Matrices to the Shift Structure of the Dirichlet Eta and Beta Functions
AU - Serge Iovleff
PY - 2026
DA - 2026/10/17
PB - HAL
DO - 10.48550/arxiv.2610.26408
UR - https://hal.science/hal-05804361v1
AB - We investigate the asymptotic behavior of the inverse of Vandermonde matrices with nodes drawn from the quadratic sequences u n = n 2 and u n = (n -1 2 ) 2 . We derive new limiting formulas for the entries of the inverses, expressed through the even and odd central factorial numbers. These formulas are then used to study the asymptotic behavior of a ratio of determinants involving a modified Vandermonde matrix. For each node sequence, the corresponding determinant ratio converges uniformly on compact subsets of the complex plane to an explicit expression involving the Dirichlet eta or beta function. We then study a family of related functions and, using an inversion theorem, exhibit their shift structure: the shifted values of the eta and beta functions are recovered as finite linear combinations of the family, which yields an integral form of these values as integrals of polynomials related to the Bernoulli and Euler polynomials against generalized (half-)Clausen functions. A unified form of the shift representation, valid for an arbitrary base point z with Re(z) > 0, brings the eta and beta families under a single theorem. On the critical line the representation is unitary and the values on the two sides are complex conjugates.
ER -