Dilcher, Karl
Permanent URI for this collectionhttps://hdl.handle.net/10222/27970
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Item type: Item , Access status: 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 type: Item , Access status: 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 type: Item , Access status: 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 type: Item , Access status: Open Access , Zeros of the Wronskian of Chebyshev and Ultraspherical Polynomials(1993-WIN 1993) Dilcher, K.; Stolarsky, K. B.No abstract available.Item type: Item , Access status: Open Access , Polynomials Related to Expansions of Certain Rational Functions in 2 Variables(1988-03) Dilcher, K.No abstract available.