Dilcher, Karlhttp://hdl.handle.net/10222/279702023-03-28T15:19:48Z2023-03-28T15:19:48ZNonlinear Identities for Bernoulli and Euler PolynomialsDilcher, Karlhttp://hdl.handle.net/10222/750642019-01-03T08:30:16Z2018-01-01T00:00:00ZNonlinear Identities for Bernoulli and Euler Polynomials
Dilcher, Karl
It is shown that a certain nonlinear expression for Bernoulli polynomials, related to higher-order convolutions, can be evaluated as a product of simple linear polynomials with integer coefficients. The proof involves higher-order Bernoulli polynomials. A similar result for Euler polynomials is also obtained, and identities for Bernoulli and Euler numbers follow as special cases.
2018-01-01T00:00:00ZDerivatives and Special Values of Higher-Order Tornheim Zeta FunctionsDilcher, KarlTomkins, Hayleyhttp://hdl.handle.net/10222/750632019-01-03T08:30:10Z2018-01-01T00:00:00ZDerivatives and Special Values of Higher-Order Tornheim Zeta Functions
Dilcher, Karl; Tomkins, Hayley
We study analytic properties of the higher-order Tornheim zeta function, defined by a certain $n$-fold series ($n\geq 2$) in $n+1$ complex variables. In particular, we consider the function $\omega_{n+1}(s)$, obtained by setting all variables equal to $s$. Using a free-parameter method due to Crandall, we first give an alternative proof of the trivial zeros of $\omega_{n+1}(s)$ and evaluate $\omega_{n+1}(0)$. Our main result, however, is the evaluation of $\omega_{n+1}'(0)$ for any $n\geq 2$. This is again achieved by using Crandall's method, and it generalizes recent results in the cases $n=2, 3$. Properties of Bernoulli numbers and of higher-order Bernoulli numbers and polynomials play an important role throughout this paper.
2018-01-01T00:00:00ZA role for generalized Fermat numbersCosgrave, John B.Dilcher, Karlhttp://hdl.handle.net/10222/714492017-12-21T16:53:48Z2016-01-01T00:00:00ZA role for generalized Fermat numbers
Cosgrave, John B.; Dilcher, Karl
We define a Gauss factorial $N_n!$ to be the product of all positive integers
up to $N$ that are relatively prime to $n\in\mathbb N$. In this paper we study
particular aspects of the Gauss factorials $\lfloor\frac{n-1}{M}\rfloor_n!$
for $M=3$ and 6, where the case of $n$ having exactly one prime factor
of the form $p\equiv 1\pmod{6}$ is of particular interest. A fundamental
role is played by those primes $p\equiv 1\pmod{3}$ with the property that the
order of $\frac{p-1}{3}!$ modulo $p$ is a power of 2 or 3 times a power of 2;
we call them Jacobi primes. Our main results are characterizations of those
$n\equiv\pm 1\pmod{M}$ of the above form that satisfy
$\lfloor\frac{n-1}{M}\rfloor_n!\equiv 1\pmod{n}$, $M=3$ or 6, in terms
of Jacobi primes and certain prime factors of generalized Fermat numbers. We
also describe the substantial and varied computations used for this paper.
Post-print version of the article, issued prior to publication.
2016-01-01T00:00:00ZZeros of the Wronskian of Chebyshev and Ultraspherical PolynomialsDilcher, K.Stolarsky, K. B.http://hdl.handle.net/10222/334212017-12-21T16:54:04Z1993-01-01T00:00:00ZZeros of the Wronskian of Chebyshev and Ultraspherical Polynomials
Dilcher, K.; Stolarsky, K. B.
No abstract available.
1993-01-01T00:00:00Z