Dilcher, Karl
Permanent URI for this collectionhttps://hdl.handle.net/10222/27970
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Item Open Access Nonlinear Identities for Bernoulli and Euler Polynomials(2018) Dilcher, KarlIt 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.Item Open Access Derivatives and Special Values of Higher-Order Tornheim Zeta Functions(2018) Dilcher, Karl; Tomkins, HayleyWe 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.Item Open Access A role for generalized Fermat numbers(American Mathematical Society, 2016) Cosgrave, John B.; Dilcher, KarlWe 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.Item Open Access Zeros of the Wronskian of Chebyshev and Ultraspherical Polynomials(1993-WIN 1993) Dilcher, K.; Stolarsky, K. B.No abstract available.Item Open Access Polynomials Related to Expansions of Certain Rational Functions in 2 Variables(1988-03) Dilcher, K.No abstract available.