Legendre-Gould Hopper-Based Sheffer Polynomials and Operational Methods

Khan, Nabiullah; Aman, Mohd; Usman, Talha; Choi, Junesang

doi:10.3390/sym12122051

Cited by 10 publications

(6 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…They have allowed the study of the relevant properties by straightforward means, paved the way to the introduction of generalized forms along with the tools to develop the underlying theoretical background. The methods suggested in [19] have given rise to a series of speculations and conjectures, and were then proved on a more solid basis in subsequent research [20][21][22][23].…”

Section: Umbral Methods and Binomial Harmonic Numbersmentioning

confidence: 99%

Inverse Derivative Operator and Umbral Methods for the Harmonic Numbers and Telescopic Series Study

2021

View full text Add to dashboard Cite

The formalism of differ-integral calculus, initially developed to treat differential operators of fractional order, realizes a complete symmetry between differential and integral operators. This possibility has opened new and interesting scenarios, once extended to positive and negative order derivatives. The associated rules offer an elegant, yet powerful, tool to deal with integral operators, viewed as derivatives of order-1. Although it is well known that the integration is the inverse of the derivative operation, the aforementioned rules offer a new mean to obtain either an explicit iteration of the integration by parts or a general formula to obtain the primitive of any infinitely differentiable function. We show that the method provides an unexpected link with generalized telescoping series, yields new useful tools for the relevant treatment, and allows a practically unexhausted tool to derive identities involving harmonic numbers and the associated generalized forms. It is eventually shown that embedding the differ-integral point of view with techniques of umbral algebraic nature offers a new insight into, and the possibility of, establishing a new and more powerful formalism.

show abstract

Section: Umbral Methods and Binomial Harmonic Numbersmentioning

confidence: 99%

Inverse Derivative Operator and Umbral Methods for the Harmonic Numbers and Telescopic Series Study

2021

View full text Add to dashboard Cite

show abstract

“…In this section, we try to provide some new identities which are not given in the previous sections by using the theory of umbral calculus (see, e.g., [4,[44][45][46]). For our purpose, some notations with modified ones and certain known facts are recalled and introduced.…”

Section: Certain Identities From Umbral Calculusmentioning

confidence: 99%

“…for arbitrary (real or complex) parameter α. A remarkably large number of polynomials, their extensions, and variants have been presented and investigated due mainly to their usefulness and applications in diverse ways in a wide range of research subjects in science and engineering such as numerical analysis, operator theory, special functions, complex analysis, statistics, sorting, and data compression (see, e.g., [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and the references therein). Take a few instances, Bell [2] proposed and analyzed so-called exponential polynomials, which are polynomials generated by di erentiating exponential functions exp(f(x)) or by expanding the exponential into a power series in x. ese polynomials (also called Bell polynomials) are related to Stirling and Bell numbers and arise in many applications.…”

Section: Introductionmentioning

confidence: 99%

Two New Generalizations of Extended Bernoulli Polynomials and Numbers, and Umbral Calculus

Khan

Ghayasuddin

Kim

et al. 2022

Journal of Mathematics

Self Cite

View full text Add to dashboard Cite

Among a remarkably large number of various extensions of polynomials and numbers, and diverse introductions of new polynomials and numbers, in this paper, we choose to introduce two new generalizations of some extended Bernoulli polynomials and numbers by using the Mittag–Leffler function and the confluent hypergeometric function. Then, we investigate certain properties and formulas of these newly introduced polynomials and numbers such as explicit representations, addition formulas, integral formulas, differential formulas, inequalities, and inequalities involving their integrals. Also, by using the theory of umbral calculus, five additional formulas regarding these new polynomials are provided. Furthermore, we propose to introduce four generalizations of the extended Euler and Genocchi polynomials. Finally, three natural problems are poised.

show abstract

“…Numerous polynomials, numbers, their extensions, degenerations, and new polynomials and new numbers have been developed and studied, owing primarily to their potential applications and use in a diverse variety of research fields (see, e.g., [66][67][68][69][70][71] and the references therein). For example, Bernoulli polynomials and numbers are among most important and useful ones (see, e.g., [5], pp.…”

Section: Sequences Of New Numbersmentioning

confidence: 99%

Reduction Formulas for Generalized Hypergeometric Series Associated with New Sequences and Applications

et al. 2021

Self Cite

View full text Add to dashboard Cite

In this paper, by introducing two sequences of new numbers and their derivatives, which are closely related to the Stirling numbers of the first kind, and choosing to employ six known generalized Kummer’s summation formulas for 2F1(−1) and 2F1(1/2), we establish six classes of generalized summation formulas for p+2Fp+1 with arguments −1 and 1/2 for any positive integer p. Next, by differentiating both sides of six chosen formulas presented here with respect to a specific parameter, among numerous ones, we demonstrate six identities in connection with finite sums of 4F3(−1) and 4F3(1/2). Further, we choose to give simple particular identities of some formulas presented here. We conclude this paper by highlighting a potential use of the newly presented numbers and posing some problems.

show abstract

Legendre-Gould Hopper-Based Sheffer Polynomials and Operational Methods

Cited by 10 publications

References 24 publications

Inverse Derivative Operator and Umbral Methods for the Harmonic Numbers and Telescopic Series Study

Inverse Derivative Operator and Umbral Methods for the Harmonic Numbers and Telescopic Series Study

Two New Generalizations of Extended Bernoulli Polynomials and Numbers, and Umbral Calculus

Reduction Formulas for Generalized Hypergeometric Series Associated with New Sequences and Applications

Contact Info

Product

Resources

About