scholarly journals Explicit Evaluation of Some Quadratic Euler-Type Sums Containing Double-Index Harmonic Numbers

2020 ◽  
Vol 77 (1) ◽  
pp. 73-98
Author(s):  
Seán Mark Stewart
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Function Approach ◽  
Harmonic Numbers ◽  
Explicit Evaluation ◽  
Double Index ◽  
Explicit Expressions

AbstractIn this paper a number of new explicit expressions for quadratic Euler-type sums containing double-index harmonic numbers H2n are given. These are obtained using ordinary generating functions containing the square of the harmonic numbers Hn. As a by-product of the generating function approach used new proofs for the remarkable quadratic series of Au-Yeung \sum\limits_{n = 1}^\infty {{{\left( {{{{H_n}} \over n}} \right)}^2} = {{17{\pi ^4}} \over {360}}} together with its closely related alternating cousin are given. New proofs for other closely related quadratic Euler-type sums that are known in the literature are also obtained.

scholarly journals Labels distance in bucket recursive trees with variable capacities of buckets

2021 ◽  
Vol 13 (2) ◽  
pp. 413-426
Author(s):  
S. Naderi ◽  
R. Kazemi ◽  
M. H. Behzadi
Keyword(s):  
Partial Differential Equations ◽  
Probability Distribution ◽  
Differential Equations ◽  
Generating Function ◽  
Generating Functions ◽  
Function Approach ◽  
Closed Formula ◽  
Tree Model ◽  
Partial Differential ◽  
Recursive Trees

Abstract The bucket recursive tree is a natural multivariate structure. In this paper, we apply a trivariate generating function approach for studying of the depth and distance quantities in this tree model with variable bucket capacities and give a closed formula for the probability distribution, the expectation and the variance. We show as j → ∞, lim-iting distributions are Gaussian. The results are obtained by presenting partial differential equations for moment generating functions and solving them.

SEQUENCE GENERATING FUNCTIONS AND THE MULTIFRACTAL FORMALISM: APPLICATIONS TO BIOPOLYMERS

Fractals ◽  
1995 ◽  
Vol 03 (01) ◽  
pp. 9-22
Author(s):  
T. GREGORY DEWEY
Keyword(s):  
Transfer Matrix ◽  
Generating Function ◽  
Generating Functions ◽  
Binary Tree ◽  
Double Helix ◽  
Function Approach ◽  
Long Term Memory ◽  
Term Memory ◽  
One Step

Helix-coil configurations in biopolymers can be represented as a symbolic sequence of helix and coil units. A given configuration can be generated by progressing down the branches of a binary tree. The probability weighting of the branches in the tree will depend on the specific model under consideration. The multifractal nature of this binary tree can be described by an adaptation of a statistical mechanical formalism that uses sequence generating functions. It is seen that the sequence generating function approach is equivalent to transfer matrix methods for processes involving one-step memory. However, this new approach can have advantages over the transfer matrix for processes involving long term memory. Using this approach, the multifractal character of helix-coil transitions in two biopolymer models is explored. The Bragg-Zimm model of the alpha helix is analyzed and is seen to be equivalent in the transfer matrix approach to a 2×2 P model with one-step memory. The perfect matched double helix model is also explored. This model shows long range effects as a result of the entropy of loop formation. Because of this entropic effect, probabilities of continuous coil sequences decay with length slower than exponentially. This gives the model similar features to those observed in intermittent chaotic systems. These effects are readily handled within the sequence generating function formalism. The double helix model shows a phase transition and this is manifested as a discontinuity in the generalized dimension. For such models with long term memory, the sequence generating function approach provides a convenient formalism for exploring multifractal behavior.

scholarly journals Summation formulae involving multiple harmonic numbers

2020 ◽  
pp. 26-26
Author(s):  
Dongwei Guo ◽  
Wenchang Chu
Keyword(s):  
Generating Function ◽  
Function Approach ◽  
Harmonic Numbers ◽  
Summation Formulae

By means of the generating function approach, we derive several summation formulae involving multiple harmonic numbers Hn,? (?), as well as other combinatorial numbers named after Bernoulli, Euler, Bell, Genocchi and Stirling.

scholarly journals A Note on Irreducible Maps with Several Boundaries

10.37236/3443 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
J. Bouttier ◽  
E. Guitter
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Planar Maps ◽  
Number Of Faces ◽  
Multiple Edges ◽  
Explicit Expressions

We derive a formula for the generating function of $d$-irreducible bipartite planar maps with several boundaries, i.e. having several marked faces of controlled degrees. It extends a formula due to Collet and Fusy for the case of arbitrary (non necessarily irreducible) bipartite planar maps, which is recovered by taking $d=0$. As an application, we obtain an expression for the number of $d$-irreducible bipartite planar maps with a prescribed number of faces of each allowed degree. Very explicit expressions are given in the case of maps without multiple edges ($d=2$), $4$-irreducible maps and maps of girth at least $6$ ($d=4$). Our derivation is based on a tree interpretation of the various encountered generating functions.

scholarly journals A General Family of q-Hypergeometric Polynomials and Associated Generating Functions

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1161
Author(s):  
Hari Mohan Srivastava ◽  
Sama Arjika
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Additional Parameter ◽  
Physical Sciences ◽  
General Family ◽  
Hypergeometric Polynomials

Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and the basic (or q-) hypergeometric polynomials are studied extensively and widely due mainly to their potential for applications in many areas of mathematical and physical sciences. Here, in this paper, we introduce a general family of q-hypergeometric polynomials and investigate several q-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of q-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized q-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various q-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called (p,q)-variations of the q-results, which we have investigated here, because the additional parameter p is obviously redundant.

On the circle polynomials of Zernike and related orthogonal sets

1954 ◽  
Vol 50 (1) ◽  
pp. 40-48 ◽  
Author(s):  
A. B. Bhatia ◽  
E. Wolf
Keyword(s):  
Unit Circle ◽  
Generating Functions ◽  
Legendre Polynomials ◽  
Phase Contrast ◽  
Diffraction Theory ◽  
Zernike Polynomials ◽  
Optical Aberrations ◽  
Orthogonal Set ◽  
Complete Orthogonal Set ◽  
Explicit Expressions

ABSTRACTThe paper is concerned with the construction of polynomials in two variables, which form a complete orthogonal set for the interior of the unit circle and which are ‘invariant in form’ with respect to rotations of axes about the origin of coordinates. It is found that though there exist an infinity of such sets there is only one set which in addition has certain simple properties strictly analogous to that of Legendre polynomials. This set is found to be identical with the set of the circle polynomials of Zernike which play an important part in the theory of phase contrast and in the Nijboer-Zernike diffraction theory of optical aberrations.The results make it possible to derive explicit expressions for the Zernike polynomials in a simple, systematic manner. The method employed may also be used to derive other orthogonal sets. One new set is investigated, and the generating functions for this set and for the Zernike polynomials are also given.

scholarly journals Generating Functions for Certain Weighted Cranks

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Shreejit Bandyopadhyay ◽  
Ae Yee
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Colored Partitions ◽  
Unimodal Sequences

Recently, George Beck posed many interesting partition problems considering the number of ones in partitions. In this paper, we first consider the crank generating function weighted by the number of ones and obtain analytic formulas for this weighted crank function under conditions of the crank being less than or equal to some specific integer. We connect these cumulative and point crank functions to the generating functions of partitions with certain sizes of Durfee rectangles. We then consider a generalization of the crank for $k$-colored partitions, which was first introduced by Fu and Tang, and investigate the corresponding generating function for this crank weighted by the number of parts in the first subpartition of a $k$-colored partition. We show that the cumulative generating functions are the same as the generating functions for certain unimodal sequences.

COUNTING NUMERICAL SEMIGROUPS WITH SHORT GENERATING FUNCTIONS

2011 ◽  
Vol 21 (07) ◽  
pp. 1217-1235 ◽  
Author(s):  
VÍCTOR BLANCO ◽  
PEDRO A. GARCÍA-SÁNCHEZ ◽  
JUSTO PUERTO
Keyword(s):  
Generating Function ◽  
Polynomial Time ◽  
Generating Functions ◽  
Time Complexity ◽  
Frobenius Number ◽  
Numerical Semigroups ◽  
Polynomial Time Complexity ◽  
Theoretical Results

This paper presents a new methodology to compute the number of numerical semigroups of given genus or Frobenius number. We apply generating function tools to the bounded polyhedron that classifies the semigroups with given genus (or Frobenius number) and multiplicity. First, we give theoretical results about the polynomial-time complexity of counting these semigroups. We also illustrate the methodology analyzing the cases of multiplicity 3 and 4 where some formulas for the number of numerical semigroups for any genus and Frobenius number are obtained.

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