Applications Related to the Generalized Seidel Matrix

Abstract

Let \(\alpha, \beta\) be any numbers. Given an initial sequence \(a_{0,m}\) (\(m = 0,1,2,\ldots\)), define the sequences \(a_{n,m}\) (\(n \geq 1\)) recursively by

\[a_{n,m} = \alpha a_{n-1,m} + \beta a_{n-1,m+1}; \quad \text{for n} \geq 1, m \geq 0.\]

Let \(\alpha, \beta\) be any numbers. Given an initial sequence \(a_{0,m}\) (\(m = 0,1,2,\ldots\)), define the sequences \(a_{n,m}\) (\(n \geq 1\)) recursively by

\[a_{n,m} = \alpha a_{n-1,m} + \beta a_{n-1,m+1}; \quad \text{for n} \geq 1, m \geq 0.\]

We call the matrix \((a_{n,m})_{n,m\geq 0}\) an generalized Seidel matrix with a parameter pair \((\alpha, \beta)\). If \(\alpha = \beta = 1\), then this matrix is the classical Seidel matrix. For various different parameter pairs \((\alpha, \beta)\) we will impose some evenness or oddness conditions on the exponential generating functions of the initial sequence \(a_{0,m}\) and the final sequence \(a_{n,0}\) of a generalized Seidel matrix (i.e., we require that these generating functions or certain related functions are even or odd). These conditions imply that the initial sequences and final sequences are equal to well-known classical sequences such as those of the Euler numbers, the Genocchi numbers, and the Springer numbers.

As applications, we give a straightforward proof of the continued fraction representations of the ordinary generating functions of the sequence of Genocchi numbers. And we also get the continued fractions representations of the ordinary generating functions of the Genocchi polynomials, Bernoulli polynomials, and Euler polynomials. Lastly, we give some applications of congruences for the Euler polynomials.