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Mark W. Coffey

Publications and source records attributed to Mark W. Coffey.

At least 19 recordsLinked to original sources
arXiv2020

Binomial Polynomials mimicking Riemann's Zeta Function

The (generalised) Mellin transforms of certain Chebyshev and Gegenbauer functions based upon the Chebyshev and Gegenbauer polynomials, have polynomial factors $p_n(s)$, whose zeros lie all on the `critical line' $\Re\,s=1/2$ or on the real axis (called critical polynomials). The transforms are identified in terms of combinatorial sums related to H. W. Gould's S:4/3, S:4/2 and S:3/1 binomial coefficient forms. Their `critical polynomial' factors are then identified as variants of the S:4/1 form, and more compactly in terms of certain $_3F_2(1)$ hypergeometric functions. Furthermore, we extend these results to a $1$-parameter family of polynomials with zeros only on the critical line. These polynomials possess the functional equation $p_n(s;β)=\pm p_n(1-s;β)$, similar to that for the Riemann xi function. It is shown that via manipulation of the binomial factors, these `critical polynomials' can be simplified to an S:3/2 form, which after normalisation yields the rational function $q_n(s).$ The denominator of the rational form has singularities on the negative real axis, and so $q_n(s)$ has the same `critical zeros' as the `critical polynomial' $p_n(s)$. Moreover as $s\rightarrow \infty$ along the positive real axis, $q_n(s)\rightarrow 1$ from below, mimicking $1/ζ(s)$ on the positive real line. In the case of the Chebyshev parameters we deduce the simpler S:2/1 binomial form, and with $\mathcal{C}_n$ the $n$th Catalan number, $s$ an integer, we show that polynomials $4\mathcal{C}_{n-1}p_{2n}(s)$ and $\mathcal{C}_{n}p_{2n+1}(s)$ yield integers with only odd prime factors. The results touch on analytic number theory, special function theory, and combinatorics.

math.NT↗
arXiv2019

Some integrals of the Dedekind $η$ function

Let $η$ be the weight $1/2$ Dedekind function. A unification and generalization of the integrals $\int_0^\infty f(x)η^n(ix)dx$, $n=1,3$, of Glasser \cite{glasser2009} is presented. Simple integral inequalities as well as some $n=2$, $4$, $6$, $8$, $9$, and $14$ examples are also given. A prominent result is that $$\int_0^\infty η^6 (ix)dx= \int_0^\infty xη^6 (ix)dx ={1 \over {8π}}\left({{Γ(1/4)} \over {Γ(3/4)}}\right)^2,$$ where $Γ$ is the Gamma function. The integral $\int_0^1 x^{-1} \ln x ~η(ix)dx$ is evaluated in terms of a reducible difference of pairs of the first Stieltjes constant $γ_1(a)$.

math.NT↗
arXiv2018

Generalizations of Russell-style integrals

First some definite integrals of W. H. L. Russell, almost all with trigonometric function integrands, are derived, and many generalized. Then a list is given in Russell-style of generalizations of integral identities of Amdeberhan and Moll. We conclude with a brief and noncomprehensive description of directions for further investigation, including the significant generalization to elliptic functions.

math.CA↗
arXiv2018

Integrals in Gradshteyn and Ryzhik: Hyperbolic and trigonometric function

The well known table of Gradshteyn and Ryzhik contains indefinite and definite integrals of both elementary and special functions. We give proofs of several entries containing integrands with some combination of hyperbolic and trigonometric functions. In fact, we occasionally present an extension of such entries or else give alternative evaluations. We develop connections with special cases of special functions including the Hurwitz zeta function. Before concluding we mention new integrals coming from the investigation of certain elliptic functions.

math.CA↗
arXiv2017

Adiabatic quantum computing solution of the knapsack problem

We illustrate the adiabatic quantum computing solution of the knapsack problem with both integer profits and weights. For problems with $n$ objects (or items) and integer capacity $c$, we give specific examples using both an Ising class problem Hamiltonian requiring $n+c$ qubits and a much more efficient one using $n+[\log_2 c]+1$ qubits. The discussion includes a brief mention of classical algorithms for knapsack, applications of this commonly occurring problem, and the relevance of further studies both theoretically and numerically of the behavior of the energy gap. Included too is a demonstration and commentary on a version of quantum search using a certain Ising model. Furthermore, an Appendix presents analytic results concerning the boundary for the easy-versus-hard problem-instance phase transition for the special case subset sum problem.

quant-ph↗
arXiv2017

Summatory relations and prime products for the Stieltjes constants, and other related results

The Stieltjes constants $γ_k(a)$ appear in the regular part of the Laurent expansion for the Hurwitz zeta function $ζ(s,a)$. We present summatory results for these constants $γ_k(a)$ in terms of fundamental mathematical constants such as the Catalan constant, and further relate them to products of rational functions of prime numbers. We provide examples of infinite series of differences of Stieltjes constants evaluating as volumes in hyperbolic $3$-space. We present a new series representation for the difference of the first Stieltjes constant at rational arguments. We obtain expressions for $ζ(1/2)L_{-p}(1/2)$, where for primes $p>7$, $L_{-p}(s)$ are certain $L$-series, and remarkably tight bounds for the value $ζ(1/2)$, $ζ(s)=ζ(s,1)$ being the Riemann zeta function.

math.NT↗
arXiv2016

On Higher Dimensional Fibonacci Numbers, Chebyshev Polynomials and Sequences of Vector Convergents

We study higher-dimensional interlacing Fibonacci sequences, generated via both Chebyshev type functions and $m$-dimensional recurrence relations. For each integer $m$, there exist both rational and integer versions of these sequences, where the underlying prime congruence structures of the rational sequence denominators enables the integer sequence to be recovered. From either the rational or the integer sequences we construct sequences of vectors in $\mathbb{Q}^m$, which converge to irrational algebraic points in $\mathbb{R}^m$. The rational sequence terms can be expressed as simple recurrences, trigonometric sums, binomial polynomials, sums of squares, and as sums over ratios of powers of the signed diagonals of the regular unit $n$-gon. These sequences also exhibit a "rainbow type" quality, and correspond to the Fleck numbers at negative indices, leading to some combinatorial identities involving binomial coefficients. It is shown that the families of orthogonal generating polynomials defining the recurrence relations employed, are divisible by the minimal polynomials of certain algebraic numbers, and the three-term recurrences and differential equations for these polynomials are derived. Further results relating to the Christoffel-Darboux formula, Rodrigues' formula and raising and lowering operators are also discussed. Moreover, it is shown that the Mellin transforms of these polynomials satisfy a functional equation of the form $p_n(s)=\pm p_n(1-s)$, and have zeros only on the critical line Re $s=1/2$.

math.NT↗
arXiv2016

Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers

The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, are generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of double zeta values. A set of identities for the Ramanujan and generalized Ramanujan polynomials is presented. An alternative proof of Lettington's identity is provided, together with its generalizations to the Hurwitz and Lerch zeta functions, hence to Dirichlet $L$ series, to Eisenstein series, and to general Mellin transforms. The Hurwitz numbers $\tilde{H}_n$ occur in the Laurent expansion about the origin of a certain Weierstrass $\wp$ function for a square lattice, and are highly analogous to the Bernoulli numbers. An integral representation of the Laurent coefficients about the origin for general $\wp$ functions, and for these numbers in particular, is presented. As a Corollary, the asymptotic form of the Hurwitz numbers is determined. In addition, a series representation of the Hurwitz numbers is given, as well as a new recurrence.

math.NT↗
arXiv2016

Note on the Stieltjes constants: series with Stirling numbers of the first kind

The Stieltjes constants $γ_k(a)$ appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function $ζ(s,a)$ about $s=1$. We generalize the integral and Stirling number series results of [4] for $γ_k(a=1)$. Along the way, we point out another recent asymptotic development for $γ_k(a)$ which provides convenient and accurate results for even modest values of $k$.

math.NT↗
arXiv2015

Reductions of particular hypergeometric functions $_3F_2(a,a+1/3,a+2/3;p/3,q/3;\pm 1)$

We principally present reductions of certain generalized hypergeometric functions $_3F_2(\pm 1)$ in terms of products of elementary functions. Most of these results have been known for some time, but one of the methods, wherein we simultaneously solve for three alternating binomial sums, may be new. We obtain a functional equation holding for all three of this set of alternating binomial sums. Using successive derivatives, we show how related chains of $_3F_2(\pm 1)$ values may be obtained. It may be emphasized that we make no reliance on the WZ method for hypergeometric summation. Additional material on Pochhammer symbols and certain of their products is presented in an Appendix to supplement the pedagogical content of the paper.

math.CA↗
arXiv2015

Generalized raising and lowering operators for supersymmetric quantum mechanics

Supersymmetric quantum mechanics has many applications, and typically uses a raising and lowering operator formalism. For one dimensional problems, we show how such raising and lowering operators may be generalized to include an arbitrary function. As a result, the usual Rodrigues' formula of the theory of orthogonal polynomials may be recovered in special cases, and it may otherwise be generalized to incorporate an arbitrary function. We provide example generalized operators for several important classical orthogonal polynomials, including Chebyshev, Gegenbauer, and other polynomials. In particular, as concerns Legendre polynomials and associated Legendre functions, we supplement and generalize results of Bazeia and Das.

math-ph↗
arXiv2014

Analytic results for spatial coherence and information entropy of an optical vortex field

Optical vortex fields have applications in information processing and storage and in the manipulation of microscopic particles. We present analytic results for quantities describing the extent of spatial coherence and entropy of one-dimensional projections of a vortex field. Sums of squares of values of certain Jacobi polynomials are important in the analysis. A family of summation identities is presented that provides the moments of an associated discrete probability distribution.

math-ph↗
arXiv2014

Functional equations for the Stieltjes constants

The Stieltjes constants $γ_k(a)$ appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function $ζ(s,a)$ about $s=1$. We present the evaluation of $γ_1(a)$ and $γ_2(a)$ at rational argument, being of interest to theoretical and computational analytic number theory and elsewhere. We give multiplication formulas for $γ_0(a)$, $γ_1(a)$, and $γ_2(a)$, and point out that these formulas are cases of an addition formula previously presented. We present certain integral evaluations generalizing Gauss' formula for the digamma function at rational argument. In addition, we give the asymptotic form of $γ_k(a)$ as $a \to 0$ as well as a novel technique for evaluating integrals with integrands with $\ln(-\ln x)$ and rational factors.

math.CV↗
arXiv2013

Mellin transforms with only critical zeros: Legendre functions

We consider the Mellin transforms of certain Legendre functions based upon the ordinary and associated Legendre polynomials. We show that the transforms have polynomial factors whose zeros lie all on the critical line Re $s=1/2$. The polynomials with zeros only on the critical line are identified in terms of certain $_3F_2(1)$ hypergeometric functions. These polynomials possess the functional equation $p_n(s)=(-1)^{\lfloor n/2 \rfloor} p_n(1-s)$. Other hypergeometric representations are presented, as well as certain Mellin transforms of fractional part and fractional part-integer part functions. The results should be of interest to special function theory, combinatorial geometry, and analytic number theory.

math-ph↗
arXiv2013

Mellin transforms with only critical zeros: Chebyshev and Gegenbauer functions

We consider the Mellin transforms of certain Chebyshev functions based upon the Chebyshev polynomials. We show that the transforms have polynomial factors whose zeros lie all on the critical line or on the real line. The polynomials with zeros only on the critical line are identified in terms of certain $_3F_2(1)$ hypergeometric functions. Furthermore, we extend this result to a 1-parameter family of polynomials with zeros only on the critical line. These polynomials possess the functional equation $p_n(s;β)=(-1)^{\lfloor n/2 \rfloor} p_n(1-s;β)$. We then present the generalization to the Mellin transform of certain Gegenbauer functions. The results should be of interest to special function theory, combinatorics, and analytic number theory.

math-ph↗
arXiv2013

Mellin transforms with only critical zeros: generalized Hermite functions

We consider the Mellin transforms of certain generalized Hermite functions based upon certain generalized Hermite polynomials, characterized by a parameter $μ>-1/2$. We show that the transforms have polynomial factors whose zeros lie all on the critical line. The polynomials with zeros only on the critical line are identified in terms of certain $_2F_1(2)$ hypergeometric functions, being certain scaled and shifted Meixner-Pollaczek polynomials. Other results of special function theory are presented.

math.CV↗
arXiv2013

The Zagier polynomials. Part II: Arithmetic properties of coefficients

The modified Bernoulli numbers \begin{equation*} B_{n}^{*} = \sum_{r=0}^{n} \binom{n+r}{2r} \frac{B_{r}}{n+r}, \quad n > 0 \end{equation*} introduced by D. Zagier in 1998 were recently extended to the polynomial case by replacing $B_{r}$ by the Bernoulli polynomials $B_{r}(x)$. Arithmetic properties of the coefficients of these polynomials are established. In particular, the 2-adic valuation of the modified Bernoulli numbers is determined. A variety of analytic, umbral, and asymptotic methods is used to analyze these polynomials.

math.NT↗