scholarly journals Generating Functions and Generalized Dedekind Sums

10.37236/1326 ◽  
1996 ◽  
Vol 4 (2) ◽  
Author(s):  
Ira Gessel
Keyword(s):  
Rational Function ◽  
Explicit Formula ◽  
Simple Form ◽  
Generating Functions ◽  
Dedekind Sums ◽  
Roots Of Unity ◽  
Partial Fractions

We study sums of the form $\sum_\zeta R(\zeta)$, where $R$ is a rational function and the sum is over all $n$th roots of unity $\zeta$ (often with $\zeta =1$ excluded). We call these generalized Dedekind sums, since the most well-known sums of this form are Dedekind sums. We discuss three methods for evaluating such sums: The method of factorization applies if we have an explicit formula for $\prod_\zeta (1-xR(\zeta))$. Multisection can be used to evaluate some simple, but important sums. Finally, the method of partial fractions reduces the evaluation of arbitrary generalized Dedekind sums to those of a very simple form.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

scholarly journals Miscellaneous Properties of the Gamma Distribution Polynomials

2019 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz
Keyword(s):  
Integral Representation ◽  
Explicit Formula ◽  
Gamma Distribution ◽  
Generating Functions ◽  
Euler Polynomials ◽  
Addition Formula ◽  
Special Polynomial

The main aim of this paper is to investigate multifarious properties and relations for the gamma distribution. The approach to reach this purpose will be introducing a special polynomial including gamma distribution. Several formulas covering addition formula, derivative property, integral representation and explicit formula are derived by means of the series manipulation method. Furthermore, two correlations including Bernoulli and Euler polynomials for gamma distribution polynomials are provided by utilizing of their generating functions.

scholarly journals Explicit Formulas for the First Form (q,r)-Dowling Numbers and (q,r)-Whitney-Lah Numbers

2021 ◽  
Vol 14 (1) ◽  
pp. 65-81
Author(s):  
Roberto Bagsarsa Corcino ◽  
Jay Ontolan ◽  
Maria Rowena Lobrigas
Keyword(s):  
Recurrence Relation ◽  
Explicit Formula ◽  
Taylor Series ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Difference Operator ◽  
Interpolation Formula ◽  
Explicit Formulas ◽  
Fundamental Properties ◽  
Whitney Numbers

In this paper, a q-analogue of r-Whitney-Lah numbers, also known as (q,r)-Whitney-Lah number, denoted by $L_{m,r} [n, k]_q$ is defined using the triangular recurrence relation. Several fundamental properties for the q-analogue are established such as vertical and horizontal recurrence relations, horizontal and exponential generating functions. Moreover, an explicit formula for (q, r)-Whitney-Lah number is derived using the concept of q-difference operator, particularly, the q-analogue of Newton’s Interpolation Formula (the umbral version of Taylor series). Furthermore, an explicit formula for the first form (q, r)-Dowling numbers is obtained which is expressed in terms of (q,r)-Whitney-Lah numbers and (q,r)-Whitney numbers of the second kind.

PARTITIONS WITH AN ARBITRARY NUMBER OF SPECIFIED DISTANCES

2019 ◽  
Vol 101 (1) ◽  
pp. 35-39 ◽  
Author(s):  
BERNARD L. S. LIN
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Generating Functions ◽  
Arbitrary Number ◽  
Positive Integers

For positive integers $t_{1},\ldots ,t_{k}$, let $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ (respectively $p(n,t_{1},t_{2},\ldots ,t_{k})$) be the number of partitions of $n$ such that, if $m$ is the smallest part, then each of $m+t_{1},m+t_{1}+t_{2},\ldots ,m+t_{1}+t_{2}+\cdots +t_{k-1}$ appears as a part and the largest part is at most (respectively equal to) $m+t_{1}+t_{2}+\cdots +t_{k}$. Andrews et al. [‘Partitions with fixed differences between largest and smallest parts’, Proc. Amer. Math. Soc.143 (2015), 4283–4289] found an explicit formula for the generating function of $p(n,t_{1},t_{2},\ldots ,t_{k})$. We establish a $q$-series identity from which the formulae for the generating functions of $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ and $p(n,t_{1},t_{2},\ldots ,t_{k})$ can be obtained.

scholarly journals NON-ABELIAN REIDEMEISTER TORSION FOR TWIST KNOTS

2009 ◽  
Vol 18 (03) ◽  
pp. 303-341 ◽  
Author(s):  
JÉRÔME DUBOIS ◽  
VU HUYNH ◽  
YOSHIKAZU YAMAGUCHI
Keyword(s):  
Rational Function ◽  
Explicit Formula ◽  
Reidemeister Torsion ◽  
Cusp Shape

This paper gives an explicit formula for the SL2(ℂ)-non-abelian Reidemeister torsion as defined in [6] in the case of twist knots. For hyperbolic twist knots, we also prove that the non-abelian Reidemeister torsion at the holonomy representation can be expressed as a rational function evaluated at the cusp shape of the knot.

An Algorithm for Partial-Fraction Expansion

1972 ◽  
Vol 65 (3) ◽  
pp. 237-239
Author(s):  
Joseph W. Rogers ◽  
Margaret Anne Rogers
Keyword(s):  
Rational Function ◽  
Partial Fraction ◽  
Partial Fraction Expansion ◽  
Partial Fractions

We Usually expand a rational function by first reducing it to the sum of a polynomial and a proper fraction that is the quotient of two polynomials in which the degree of the numerator is less than the degree of the denominator. We expand the fraction by writing it as a sum of partial fractions with undetermined numerator coefficients.

ON OHTSUKI'S INVARIANTS OF HOMOLOGY 3-SPHERES

1999 ◽  
Vol 08 (08) ◽  
pp. 1049-1063 ◽  
Author(s):  
RUTH J. LAWRENCE
Keyword(s):  
Power Series ◽  
Explicit Formula ◽  
Formal Power Series ◽  
Original Work ◽  
Roots Of Unity ◽  
Integral Expression ◽  
Rational Homology ◽  
Formal Power

By analysing Ohtsuki's original work in which he produced a formal power series invariant of rational homology 3-spheres, we obtain a simplified explicit formula for them, which may also be compared with Rozansky's integral expression. We further show their relation to the exact SO(3) Witten-Reshetikhin-Turaev invariants at roots of unity in a stronger form than that given in Ohtsuki's original work.

scholarly journals Dedekind sums s(a, b) and inversions modulo b

2015 ◽  
Vol 11 (08) ◽  
pp. 2325-2339
Author(s):  
Yiwang Chen ◽  
Nicholas Dunn ◽  
Campbell Hewett ◽  
Shashwat Silas
Keyword(s):  
Kloosterman Sums ◽  
Dedekind Sums ◽  
Roots Of Unity ◽  
Sufficient Condition ◽  
Dedekind Sum

We introduce the inversion polynomial for Dedekind sums fb(x) = ∑ x inv (a, b) to study the number of s(a, b) which have the same value for a given b. We prove several properties of this polynomial and present some conjectures. We also introduce connections between Kloosterman sums and the inversion polynomial evaluated at particular roots of unity. Finally, we improve on previously known bounds for the second highest value of the Dedekind sum and provide a conjecture for a possible generalization. Lastly, we include a new sufficient condition for the inequality of two Dedekind sums based on the reciprocity formula.

Staircase words and Chebyshev polynomials

2010 ◽  
Vol 4 (1) ◽  
pp. 81-95 ◽  
Author(s):  
Arnold Knopfmacher ◽  
Toufik Mansour ◽  
Augustine Munagi ◽  
Helmut Prodinger
Keyword(s):  
Explicit Formula ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Trigonometric Sums ◽  
Cyclic Words

A word ? = ?1... ?n over the alphabet [k]={1,2,...,k} is said to be a staircase if there are no two adjacent letters with difference greater than 1. A word ? is said to be staircase-cyclic if it is a staircase word and in addition satisfies |?n??1|?1. We find the explicit generating functions for the number of staircase words and staircase-cyclic words in [k]n, in terms of Chebyshev polynomials of the second kind. Additionally, we find explicit formula for the numbers themselves, as trigonometric sums. These lead to immediate asymptotic corollaries. We also enumerate staircase necklaces, which are staircase-cyclic words that are not equivalent up to rotation.

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