AN ANALOGUE OF EULER’S IDENTITY AND SPLIT PERFECT PARTITIONS

2018 ◽  
Vol 99 (03) ◽  
pp. 353-361
Author(s):  
MEGHA GOYAL
Keyword(s):  
Generating Function ◽  
New Class ◽  
Appl Math ◽  
Euler’S Identity

We give the generating function of split$(n+t)$-colour partitions and obtain an analogue of Euler’s identity for split$n$-colour partitions. We derive a combinatorial relation between the number of restricted split$n$-colour partitions and the function$\unicode[STIX]{x1D70E}_{k}(\unicode[STIX]{x1D707})=\sum _{d|\unicode[STIX]{x1D707}}d^{k}$. We introduce a new class of split perfect partitions with$d(a)$copies of each part$a$and extend the work of Agarwal and Subbarao [‘Some properties of perfect partitions’,Indian J. Pure Appl. Math 22(9) (1991), 737–743].

scholarly journals A New Class of Higher-Order Hypergeometric Bernoulli Polynomials Associated with Lagrange–Hermite Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 648
Author(s):  
Ghulam Muhiuddin ◽  
Waseem Ahmad Khan ◽  
Ugur Duran ◽  
Deena Al-Kadi
Keyword(s):  
Generating Function ◽  
Functional Equations ◽  
Laguerre Polynomials ◽  
Hermite Polynomials ◽  
Bernoulli Polynomials ◽  
Higher Order ◽  
New Class

The purpose of this paper is to construct a unified generating function involving the families of the higher-order hypergeometric Bernoulli polynomials and Lagrange–Hermite polynomials. Using the generating function and their functional equations, we investigate some properties of these polynomials. Moreover, we derive several connected formulas and relations including the Miller–Lee polynomials, the Laguerre polynomials, and the Lagrange Hermite–Miller–Lee polynomials.

scholarly journals Quasi cl-supercontinuous functions and their function spaces

2012 ◽  
Vol 45 (3) ◽  
Author(s):  
J. K. Kohli ◽  
Jeetendra Aggarwal
Keyword(s):  
Continuous Functions ◽  
Function Spaces ◽  
Strong Form ◽  
General Topology ◽  
Regular Space ◽  
Basic Properties ◽  
New Class ◽  
Mathematical Literature ◽  
Class Of Functions ◽  
Appl Math

AbstractA new class of functions called ‘quasi cl-supercontinuous functions’ is introduced. Basic properties of quasi cl-supercontinuous functions are studied and their place in the hierarchy of variants of continuity that already exist in the mathematical literature is elaborated. The notion of quasi cl-supercontinuity, in general, is independent of continuity but coincides with cl-supercontinuity (≡ clopen continuity) (Applied General Topology 8(2) (2007), 293–300; Indian J. Pure Appl. Math. 14(6) (1983), 767–772), a significantly strong form of continuity, if range is a regular space. The class of quasi cl-supercontinuous functions properly contains each of the classes of (i) quasi perfectly continuous functions and (ii) almost cl-supercontinuous functions; and is strictly contained in the class of quasi

scholarly journals Multifarious Results for q-Hermite Based Frobenius Type Eulerian Polynomials

2019 ◽  
Author(s):  
Waseem Khan ◽  
Idrees Ahmad Khan ◽  
Mehmet Acikgoz ◽  
Ugur Duran
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Series Representation ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Eulerian Polynomials ◽  
New Class ◽  
Genocchi Polynomials

In this paper, a new class of q-Hermite based Frobenius type Eulerian polynomials is introduced by means of generating function and series representation. Several fundamental formulas and recurrence relations for these polynomials are derived via different generating methods. Furthermore, diverse correlations including the q-Apostol-Bernoulli polynomials, the q-Apostol-Euler poynoomials, the q-Apostol-Genocchi polynomials and the q-Stirling numbers of the second kind are also established by means of the their generating functions.

scholarly journals Notes on $q$-Hermite Based Unified Apostol Type Polynomials

2019 ◽  
Author(s):  
Waseem Khan
Keyword(s):  
Generating Function ◽  
Recurrence Relations ◽  
Series Representation ◽  
Stirling Numbers ◽  
New Class

In this article, a new class of $q$-Hermite based unified Apostol type polynomials is introduced by means of generating function and series representation. Several important formulas and recurrence relations for these polynomials are derived via different generating methods. We also introduce $q$-analog of Stirling numbers of second kind of order $\nu$ by which we construct a relation including aforementioned polynomials.

scholarly journals Explicit formula of a new class of q-Hermite-based Apostol-type polynomials and generalization

2020 ◽  
Vol 26 (4) ◽  
pp. 93-102
Author(s):  
Mouloud Goubi ◽  
Keyword(s):  
Explicit Formula ◽  
Present Article ◽  
Generating Function ◽  
New Class

The present article deals with a recent study of a new class of q-Hermite-based Apostol-type polynomials introduced by Waseem A. Khan and Divesh Srivastava. We give their explicit formula and study a generalized class depending in any q-analog generating function.

scholarly journals The Zeta-G Class: Some Computational and Analytical Aspects

2020 ◽  
Vol 42 ◽  
pp. e111
Author(s):  
Ana Carla Percontini ◽  
Frank Gomes-Silva ◽  
Gauss Moutinho Crdeiro ◽  
Pedro Rafael Marinho
Keyword(s):  
Maximum Likelihood ◽  
Generating Function ◽  
Real Data ◽  
Data Set ◽  
R Language ◽  
New Class ◽  
Mathematical Properties ◽  
Special Cases ◽  
Computational Aspects

We define a new class of distributions with one extra shapeparameter including some special cases. We provide numerical and computational aspects of the new class. We proposefunctions using the \textsf{R} language to fit any distribution in this family to a data set. In addition, such functions are implemented efficientlyusing the library \textsf{Rcpp} that enables the incorporation of the codes \textsf{C++} in \textsf{R} automatically. Some examples are presentedfor using the implemented routines in practice. We derive some mathematical properties of this class including explicit expressionsfor the moments, generating function and mean deviations. We discuss the estimation of the model parametersby maximum likelihood and provide an application to a real data set.

ARITHMETIC PROPERTIES OF 3-REGULAR PARTITIONS IN THREE COLOURS

2021 ◽  
pp. 1-9
Author(s):  
ROBSON DA SILVA ◽  
JAMES A. SELLERS
Keyword(s):  
Generating Function ◽  
Infinite Family ◽  
Arithmetic Properties ◽  
Regular Partitions ◽  
Appl Math

Abstract Gireesh and Mahadeva Naika [‘On 3-regular partitions in 3-colors’, Indian J. Pure Appl. Math.50 (2019), 137–148] proved an infinite family of congruences modulo powers of 3 for the function $p_{\{3,3\}}(n)$ , the number of 3-regular partitions in three colours. In this paper, using elementary generating function manipulations and classical techniques, we significantly extend the list of proven arithmetic properties satisfied by $p_{\{3,3\}}(n).$

scholarly journals Unlabeled $(2+2)$-free posets, ascent sequences and pattern avoiding permutations

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Mireille Bousquet-Mélou ◽  
Anders Claesson ◽  
Mark Dukes ◽  
Sergey Kitaev
Keyword(s):  
Generating Function ◽  
Integer Sequences ◽  
Combinatorial Objects ◽  
New Class ◽  
Chord Diagrams ◽  
The Third ◽  
Finite Series ◽  
International Audience ◽  
Ascent Sequences ◽  
Pattern Avoiding Permutations

International audience We present statistic-preserving bijections between four classes of combinatorial objects. Two of them, the class of unlabeled $(\textrm{2+2})$-free posets and a certain class of chord diagrams (or involutions), already appeared in the literature, but were apparently not known to be equinumerous. The third one is a new class of pattern avoiding permutations, and the fourth one consists of certain integer sequences called $\textit{ascent sequences}$. We also determine the generating function of these classes of objects, thus recovering a non-D-finite series obtained by Zagier for chord diagrams. Finally, we characterize the ascent sequences that correspond to permutations avoiding the barred pattern $3\bar{1}52\bar{4}$, and enumerate those permutations, thus settling a conjecture of Pudwell. Nous présentons des bijections, transportant de nombreuses statistiques, entre quatre classes d'objets. Deux d'entre elles, la classe des EPO (ensembles partiellement ordonnés) sans motif $(\textrm{2+2})$ et une certaine classe d'involutions, sont déjà apparues dans la littérature. La troisième est une classe de permutations à motifs exclus, et la quatrième une classe de suites que nous appelons $\textit{suites à montées}$. Nous déterminons ensuite la série génératrice de ces classes, retrouvant ainsi un résultat prouvé par Zagier pour les involutions sus-mentionnées. La série obtenue n'est pas D-finie. Apparemment, le fait qu'elle compte aussi les EPO sans motif $(\textrm{2+2})$ est nouveau. Finalement, nous caractérisons les suites à montées qui correspondent aux permutations évitant le motif barré $3\bar{1}52\bar{4}$ et énumérons ces permutations, ce qui démontre une conjecture de Pudwell.

scholarly journals Weakly prudent self-avoiding bridges

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Axel Bacher ◽  
Nicholas Beaton
Keyword(s):  
Generating Function ◽  
Square Lattice ◽  
Growth Constant ◽  
Arbitrary Size ◽  
New Class ◽  
Advantages And Disadvantages ◽  
International Audience ◽  
Self Avoiding Walks

International audience We define and enumerate a new class of self-avoiding walks on the square lattice, which we call <i>weakly prudent bridges</i>. Their definition is inspired by two previously-considered classes of self-avoiding walks, and can be viewed as a combination of those two models. We consider several methods for recursively generating these objects, each with its own advantages and disadvantages, and use these methods to solve the generating function, obtain very long series, and randomly generate walks of arbitrary size. We find that the growth constant of these walks is approximately 2.58, which is larger than that of any previously-solved class of self-avoiding walks.

scholarly journals A New Class of Hermite-Apostol Type Frobenius-Euler Polynomials and Its Applications

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 652
Author(s):  
Serkan Araci ◽  
Mumtaz Riyasat ◽  
Shahid Wani ◽  
Subuhi Khan
Keyword(s):  
Generating Function ◽  
Zeta Functions ◽  
Euler Polynomials ◽  
Power Sums ◽  
New Class ◽  
Summation Formulae ◽  
Special Cases ◽  
Hybrid Class

The article is written with the objectives to introduce a multi-variable hybrid class, namely the Hermite–Apostol-type Frobenius–Euler polynomials, and to characterize their properties via different generating function techniques. Several explicit relations involving Hurwitz–Lerch Zeta functions and some summation formulae related to these polynomials are derived. Further, we establish certain symmetry identities involving generalized power sums and Hurwitz–Lerch Zeta functions. An operational view for these polynomials is presented, and corresponding applications are given. The illustrative special cases are also mentioned along with their generating equations.

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