The Relation of the D-Orthogonal Polynomials to the Appell Polynomials
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Asymptotic Properties of Biorthogonal Polynomials Systems Related to Hermite and Laguerre Polynomials
Asymptotic properties of biorthogonal polynomials systems related to Hermite and Laguerre polynomials Yan Xu School of Mathematics and Quantitative Economics, Center for Econometric analysis and Forecasting, Dongbei University of Finance and Economics, Liaoning, 116025, PR China Abstract In this paper, the structures to a family of biorthogonal polynomials that ap- proximate to the Hermite and Generalized Laguerre polynomials are discussed respectively. Therefore, the asymptotic relation between several orthogonal polynomials and combinatorial polynomials are derived from the systems, which in turn verify the Askey scheme of hypergeometric orthogonal polynomials. As the applications of these properties, the asymptotic representations of the gen- eralized Buchholz, Laguerre, Ultraspherical (Gegenbauer), Bernoulli, Euler, Meixner and Meixner-Pllaczekare polynomials are derived from the theorems di- rectly. The relationship between Bernoulli and Euler polynomials are shown as a special case of the characterization theorem of the Appell sequence generated by α scaling functions. Keywords: Hermite Polynomial, Laguerre Polynomial, Appell sequence, Askey Scheme, B-splines, Bernoulli Polynomial, Euler polynomials. 2010 MSC: 42C05, 33C45, 41A15, 11B68 arXiv:1503.05387v1 [math.CA] 3 Feb 2015 ✩This work is supported by the Natural Science Foundation of China (11301060), China Postdoctoral Science Foundation (2013M541234, 2014T70258) and Outstanding Scientific In- novation Talents Program of DUFE (DUFE2014R20). ∗Email: yan [email protected] Preprint submitted to Constructive Approximation June 2, 2021 1. Introduction The Hermite polynomials follow from the generating function 2 xz z ∞ Hm(x) m e − 2 = z , z C, x R (1.1) m! ∈ ∈ m=0 X which gives the Cauchy-type integral 2 m! xz z (m+1) H (x)= e − 2 z− dz. -
General Bivariate Appell Polynomials Via Matrix Calculus and Related Interpolation Hints
mathematics Article General Bivariate Appell Polynomials via Matrix Calculus and Related Interpolation Hints Francesco Aldo Costabile, Maria Italia Gualtieri and Anna Napoli * Department of Mathematics and Computer Science, University of Calabria, 87036 Rende (CS), Italy; [email protected] (F.A.C.); [email protected] (M.I.G.) * Correspondence: [email protected] Abstract: An approach to general bivariate Appell polynomials based on matrix calculus is proposed. Known and new basic results are given, such as recurrence relations, determinant forms, differential equations and other properties. Some applications to linear functional and linear interpolation are sketched. New and known examples of bivariate Appell polynomial sequences are given. Keywords: Polynomial sequences; Appell polynomials; bivariate Appell sequence 1. Introduction Appell polynomials have many applications in various disciplines: probability the- ory [1–5], number theory [6], linear recurrence [7], general linear interpolation [8–12], operators approximation theory [13–17]. In [18], P. Appell introduced a class of polynomi- als by the following equivalent conditions: fAngn2IN is an Appell sequence (An being a Citation: Costabile, F.A.; polynomial of degree n) if either Gualtieri, M.I.; Napoli, A. General 8 Bivariate Appell Polynomials via d An(x) > = nA − (x), n ≥ 1, Matrix Calculus and Related > n 1 <> dx Interpolation Hints. Mathematics 2021, A (0) = a , a 6= 0, a 2 IR, n ≥ 0, 9, 964. https://doi.org/ > n n 0 n > 10.3390/math9090964 :> A0(x) = 1, Academic Editor: Clemente Cesarano or ¥ n xt t A(t)e = ∑ An(x) , Received: 13 March 2021 n=0 n! Accepted: 23 April 2021 ¥ tk Published: 25 April 2021 where A(t) = ∑ ak , a0 6= 0, ak 2 IR, k ≥ 0. -
Some New Identities Involving Sheffer-Appell Polynomial Sequences Via Matrix Approach
SOME NEW IDENTITIES INVOLVING SHEFFER-APPELL POLYNOMIAL SEQUENCES VIA MATRIX APPROACH MOHD SHADAB, FRANCISCO MARCELLAN´ AND SAIMA JABEE∗ Abstract. In this contribution some new identities involving Sheffer-Appell polyno- mial sequences using generalized Pascal functional and Wronskian matrices are de- duced. As a direct application of them, identities involving families of polynomials as Euler, Bernoulli, Miller-Lee and Apostol-Euler polynomials, among others, are given. 1. introduction Sequences of polynomials play an important role in many problems of pure and applied mathematics in the framework of approximation theory, statistics, combinatorics and classical analysis (see, for example, [19, 22{25]). The sequence of Sheffer polynomials constitutes one of the most important family of polynomial sequences. A polynomial sequence fsn(x)gn≥0 is said to be a Sheffer polynomial sequence [6, 9, 24, 27] if its generating function has the following form: 1 X yn A(y)exH(y) = s (x) ; (1.1) n n! n=0 where A(y) = A0 + A1y + ··· ; and 2 H(y) = H1y + H2y + ··· ; with A0 6= 0 and H1 6= 0. Let us recall an alternative definition of the Sheffer polynomial sequences [24, Pg. 17]. P1 yn P1 yn Indeed, let h(y) = n=1 hn n! ; h1 6= 0; and l(y) = n=0 ln n! , l0 6= 0; be, respectively, delta series and invertible series with complex coefficients. Then there exists a unique sequence of Sheffer polynomials fsn(x)gn≥0 satisfying the orthogonality conditions k hl(y)h(y) jsn(x)i = n!δn;k 8 n; k = 0; (1.2) where δn;k is the Kronecker delta. -
A Determinantal Approach to Appell Polynomials F.A
Journal of Computational and Applied Mathematics 234 (2010) 1528–1542 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam A determinantal approach to Appell polynomials F.A. Costabile ∗, E. Longo Department of Mathematics, University of Calabria, Italy article info a b s t r a c t Article history: A new definition by means of a determinantal form for Appell (1880) [1] polynomials Received 3 July 2008 is given. General properties, some of them new, are proved by using elementary linear Received in revised form 15 January 2009 algebra tools. Finally classic and non-classic examples are considered and the coefficients, calculated by an ad hoc Mathematica code, for particular sequences of Appell polynomials MSC: are given. 11B83 ' 2010 Elsevier B.V. All rights reserved. 65F40 Keywords: Determinant Polynomial Appell 1. Introduction In 1880 [1] Appell introduced and widely studied sequences of n-degree polynomials An.x/; n D 0; 1;::: (1) satisfying the recursive relations dAn.x/ D nAn−1.x/; n D 1; 2;:::: (2) dx In particular, Appell noticed the one-to-one correspondence of the set of such sequences fAn.x/gn and the set of numerical sequences fαngn ; α0 6D 0 given by the explicit representation n n 2 n An.x/ D αn C αn−1x C αn−2x C···C α0x ; n D 0; 1;:::: (3) 1 2 Eq. (3), in particular, shows explicitly that for each n ≥ 1 An.x/ is completely determined by An−1.x/ and by the choice of the constant of integration αn. -
“Orthogonal” Polynomials
1 2 Journal of Integer Sequences, Vol. 11 (2008), 3 Article 08.5.3 47 6 23 11 Riordan Arrays, Sheffer Sequences and “Orthogonal” Polynomials Giacomo Della Riccia Dept. of Math. and Comp. Science - Research Center Norbert Wiener University of Udine Via delle Scienze 206 33100 Udine Italy [email protected] Abstract Riordan group concepts are combined with the basic properties of convolution fam- ilies of polynomials and Sheffer sequences, to establish a duality law, canonical forms n m k k+1 x−k ρ(n, m) = m c Fn−m(m), c 6= 0, and extensions ρ(x,x − k) = (−1) x c Fk(x), where the Fk(x) are polynomials in x, holding for each ρ(n, m) in a Riordan array. Ex- n amples ρ(n, m)= m Sk(x) are given, in which the Sk(x) are “orthogonal” polynomials currently found in mathematical physics and combinatorial analysis. 1 Introduction We derive from basic principles in the theory of convolution families [10] and Sheffer se- n m quences [16] of polynomials, canonical forms ρ(n, m) = m c Fn−m(m), c =6 0, and exten- sions ρ(x, x − k) = xkcx−kF (x − k)/k!=(−1)kxk+1cx−kρ (x), holding for all elements ρ k k in the Riordan group. We show in Section 3 that the extensions are of polynomial type when c = 1. In Section 2, we define transformation rules and a duality law, that will greatly n simplify algebraic manipulations. In Section 4, we give examples ρ(n, m) = m Sn−m(m), in which S (x) are “orthogonal” polynomials currently found in mathematical physics and k combinatorial analysis. -
An Infinite Dimensional Umbral Calculus
An infinite dimensional umbral calculus Dmitri Finkelshtein Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K.; e-mail: [email protected] Yuri Kondratiev Fakult¨atf¨urMathematik, Universit¨atBielefeld, 33615 Bielefeld, Germany; e-mail: [email protected] Eugene Lytvynov Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K.; e-mail: [email protected] Maria Jo~aoOliveira Departamento de Ci^enciase Tecnologia, Universidade Aberta, 1269-001 Lisbon, Por- tugal; CMAF-CIO, University of Lisbon, 1749-016 Lisbon, Portugal; e-mail: [email protected] Abstract The aim of this paper is to develop foundations of umbral calculus on the space D0 of distri- d 0 butions on R , which leads to a general theory of Sheffer polynomial sequences on D . We define a sequence of monic polynomials on D0, a polynomial sequence of binomial type, and a Sheffer sequence. We present equivalent conditions for a sequence of monic polynomials on D0 to be of binomial type or a Sheffer sequence, respectively. We also construct a lift- ing of a sequence of monic polynomials on R of binomial type to a polynomial sequence of 0 0 binomial type on D , and a lifting of a Sheffer sequence on R to a Sheffer sequence on D . Examples of lifted polynomial sequences include the falling and rising factorials on D0, Abel, Hermite, Charlier, and Laguerre polynomials on D0. Some of these polynomials have already appeared in different branches of infinite dimensional (stochastic) analysis and played there a fundamental role. arXiv:1701.04326v2 [math.FA] 5 Mar 2018 Keywords: Generating function; polynomial sequence on D0; polynomial sequence of binomial type on D0; Sheffer sequence on D0; shift-invariance; umbral calculus on D0. -
Math Student-Part-1-2017
ISSN: 0025-5742 THE MATHEMATICS STUDENT Volume 86, Numbers 1-2, January-June (2017) (Issued: May, 2017) Editor-in-Chief J. R. PATADIA EDITORS Bruce C: Berndt George E: Andrewscopy- M: Ram Murty N: K: Thakare Satya Deo Gadadhar Misra B: Sury Kaushal Verma Krishnaswami Alladi S: K: Tomar Subhash J: Bhatt L: Sunil Chandran M: M: Shikare C: S: Aravinda A: S: Vasudeva Murthy Indranil Biswas Timothy Huber T: S: S: R: K: Rao Clare D0Cruz Atul Dixit circulation for Member's not PUBLISHED BY THE INDIAN MATHEMATICAL SOCIETY www.indianmathsociety.org.in THE MATHEMATICS STUDENT Edited by J. R. PATADIA In keeping with the current periodical policy, THE MATHEMATICS STUDENT will seek to publish material of interest not just to mathematicians with specialized interest but to the postgraduate students and teachers of mathematics in India. With this in view, it will ordinarily publish material of the following type: 1. the texts (written in a way accessible to students) of the Presidential Addresses, the Plenary talks and the Award Lectures delivered at the Annual Conferences. 2. general survey articles, popular articles, expository papers, Book-Reviews. 3. problems and solutions of the problems, 4. new, clever proofs of theorems that graduate / undergraduate students might see in their course work, 5. research papers of interest to larger readership, and 6. articles that arouse curiosity and interest for learning mathematics among readers and motivate them for doing mathematics. Articles of the above type are invited for publication in THE MATHEMATICS STUDENT. Manuscripts intended for publication should be submitted online in the LATEX and .pdf file including figures and tables to the Editor J. -
Algebraic Theory of Appell Polynomials with Application to General Linear Interpolation Problem
Provisional chapter Algebraic Theory of Appell Polynomials with Application to General Linear Interpolation Problem Francesco Aldo Costabile and Elisabetta Longo Additional information is available at the end of the chapter http://dx.doi.org/10.5772/46482 1. Introduction In 1880 P. E. Appell ([1]) introduced and widely studied sequences of n-degree polynomials An(x), n = 0, 1, ... (id1) satisfying the differential relation DAn(x) = nAn-1(x), n = 1, 2, ... (id2) Sequences of polynomials, verifying the (▭), nowadays called Appell polynomials, have been well studied because of their remarkable applications not only in different branches of mathematics ([2], [3]) but also in theoretical physics and chemistry ([4], [5]). In 1936 an initial bibliography was provided by Davis (p. 25[6]). In 1939 Sheffer ([7]) introduced a new class of polynomials which extends the class of Appell polynomials; he called these polynomials of type zero, but nowadays they are called Sheffer polynomials. Sheffer also noticed the sim‐ ilarities between Appell polynomials and the umbral calculus, introduced in the second half of the 19th century with the work of such mathematicians as Sylvester, Cayley and Blissard (for examples, see [8]). The Sheffer theory is mainly based on formal power series. In 1941 Steffensen ([9]) published a theory on Sheffer polynomials based on formal power series too. However, these theories were not suitable as they did not provide sufficient computational tools. Afterwards Mullin, Roman and Rota ([10], [11], [12]), using operators method, gave a beautiful theory of umbral calculus, including Sheffer polynomials. Recently, Di Bucchianico and Loeb ([13]) summarized and documented more than five hundred old and new findings related to Appell polynomial sequences. -
Some Curious Results Involving Certain Polynomials
Available online a t www.scholarsresearchlibrary.com Scholars Research Library European Journal of Applied Engineering and Scientific Research, 2014, 3 (1):21-30 (http://scholarsresearchlibrary.com/archive.html) ISSN: 2278 – 0041 Some curious results involving certain polynomials Salahuddin and R.K. Khola Mewar University, Gangrar, Chittorgarh (Rajasthan) , India _____________________________________________________________________________________________ ABSTRACT In this paper we have developed some curious results involving certain polynomials. Key Words : Hypergeometric function, Laplace Transform, Lucas Polynomials, Gegenbaur polynomials, Harmonic number, Bernoulli Polynomial, Hermite Polynomials. 2010 MSC NO : 11B39, 11B68, 33C05, 33C45, 33D50, 33D60 _____________________________________________________________________________________________ INTRODUCTION We have the generalized Gaussian hypergeometric function of one variable ∞ ….. AFB(a 1,a 2,…,a A;b 1,b 2,…b B;z ) =∑ (1.1) ….. ! where the parameters b1 , b 2 , ….,b B are neither zero nor negative integers and A , B are non negative integers. The series converges for all finite z if A ≤ B, converges for │z│<1 if A=B +1, diverges for all z, z ≠ 0 if A > B+1. Laplace Transform The Laplace Transform of a function f(t) , defined for all real numbers t ≥0, is the function F(s), defined by : ∞ − Fs()= Lft {()}() s = ∫ est ftdt () 0 (1.2) The parameter s is a complex number: s=σ + i ω , with real number σ and ω . Jacobi Polynomials In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials ) are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight α β (1−x ) (1 + x ) on the interval [-1, 1]. The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials. -
Bernoulli-Dunkl and Euler-Dunkl Polynomials and Their Generalizations∗,†
Bernoulli-Dunkl and Euler-Dunkl polynomials and their generalizations∗,y Oscar´ Ciaurri, Judit M´ınguezCeniceros, and Juan L. Varona Departamento de Matem´aticasy Computaci´on, Universidad de La Rioja, 26006 Logro~no, Spain E-mail addresses: [email protected], [email protected], [email protected] March 13, 2019 Abstract Bernoulli-Dunkl and Euler-Dunkl polynomials are generalizations of the classical Bernoulli and Euler polynomials, using the Dunkl operator instead of the differential operator. In this paper, we study properties of these polynomials that extend some of the well-known identities in the classical case, such as the Euler-Maclaurin or the Boole summation formulas in the Dunkl context. 2010 Mathematics Subject Classification: Primary 11B68; Secondary 42C10, 33C10, 11M41. Keywords: Appell-Dunkl sequences, Bernoulli-Dunkl polynomials, Euler- Dunkl polynomials. 1 Introduction 1 An Appell sequence, fPn(x)gn=0, is defined by means of a Taylor generating expansion 1 X tn A(t)ext = P (x) ; (1.1) n n! n=0 where A(t) is an analytic function in a neighborhood around t = 0 and A(0) 6= 0. It is easy to prove that, under these circumstances, Pn(x) is a polynomial of 0 degree n and Pn(x) = nPn−1(x). Typical examples of Appell sequences are the ∗Partially supported by MTM2015-65888-C4-4-P (Ministerio de Econom´ıay Competitivi- dad) and Feder Funds (European Union). yThis paper has been published in: Rev. R. Acad. Cienc. Exactas F´ıs.Nat. Ser. A Mat. RACSAM 113 (2019), 2853{2876, https://doi.org/10.1007/s13398-019-00662-z 1 1 1 Bernoulli polynomials fBn(x)gn=0, the Euler polynomials fEn(x)gn=0, and the 1 probabilists' Hermite polynomials fHen(x)gn=0, that are defined taking A(t) = 2 t=(et −1), 2=(et +1) or e−t =2, respectively (a slight variation are the physicists' 1 −t2 2xt P1 n Hermite polynomials fHn(x)gn=0 defined by e e = n=0 Hn(x)t =n!). -
Appell and Sheffer Sequences: on Their Characterizations Through
APPELL AND SHEFFER SEQUENCES: ON THEIR CHARACTERIZATIONS THROUGH FUNCTIONALS AND EXAMPLES SERGIO A. CARRILLO AND MIGUEL HURTADO Abstract. The aim of this paper is to present a new simple recurrence for Appell and Sheffer sequences in terms of the linear functional that defines them, and to explain how this is equivalent to several well-known characterizations appearing in the literature. We also give several exam- ples, including integral representations of the inverse operators associated to Bernoulli and Euler polynomials, and a new integral representation of the re-scaled Hermite d-orthogonal polynomials generalizing the Weierstrass operator related to the Hermite polynomials. 1. Introduction A remarkable class of polynomials are the Appell sequences having applications in Number theory, Probability, and the theory of functions. They vastly generalize monomials arising naturally from Taylor’s formula with integral rest, see [9, Chapitre VI] [10, Appendix A] and they include famous polynomial sequences. For instance, the Bernoulli polynomials useful in numerical integration and asymptotic analysis –Euler-Maclaurin formula [21]–; or the Euler polynomials which lead to Euler- Boole formula [8]. On the other hand, a wide class of Appell sequences provides special values of transcendental functions, as recently proved in [19], extending the well-known case of the Bernoulli polynomials as the values at negative integers of the Hurwitz zeta function. Appell sequences are a subclass of Sheffer sequences –type zero polynomials– which were treated by I. M. Sheffer as solutions of families of differential and difference equations [26]. Later on, their study, leaded by Rota and Roman, developed in what it is known today as Umbral Calculus [23, 24, 25]. -
An Algebraic Exposition of Umbral Calculus with Application to General Linear Interpolation Problem – a Survey
PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 96 (110) (2014), 67–83 DOI: 10.2298/PIM1410067C AN ALGEBRAIC EXPOSITION OF UMBRAL CALCULUS WITH APPLICATION TO GENERAL LINEAR INTERPOLATION PROBLEM – A SURVEY Francesco Aldo Costabile and Elisabetta Longo Abstract. A systematic exposition of Sheffer polynomial sequences via de- terminantal form is given. A general linear interpolation problem related to Sheffer sequences is considered. It generalizes many known cases of linear interpolation. Numerical examples and conclusions close the paper. 1. The modern umbral calculus In the 1970s Rota and his collaborators [17,19,20] began to construct a com- pletely rigorous foundation for the classical umbral calculus, consisted primarily of a symbolic technique for the manipulation of numerical and polynomial sequences. The theory of Rota et al. was based on the relatively modern ideas of linear func- tional, linear operator and adjoint. This theory followed that less efficient of gener- ating function methods; in fact, Appell [1], Sheffer [22] and Steffensen [23] based their theories on formal power series. These theories can be criticized both for their formalism not suitable for nonspecialists and for insufficient computational tools. The umbral calculus, because of its numerous applications in many branches of mathematics, physics, chemistry and engineering [24], has received many atten- tions from researchers. Recently, Di Bucchianico and Loeb [14] summarized and documented more than five hundred old and new findings related to umbral calcu- lus. In last years attention has centered on finding novel approaches. For example, in [21] the connection between Sheffer polynomials and Riordan array is sketched and in [16] the isomorphism between the Sheffer group and the Riordan Group is proved.