scholarly journals A Triple Lacunary Generating Function for Hermite Polynomials

10.37236/1927 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Ira M. Gessel ◽  
Pallavi Jayawant
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Function ◽  
Generating Functions ◽  
Hermite Polynomials ◽  
Combinatorial Proof ◽  
Classical Orthogonal Polynomials ◽  
Combinatorial Objects ◽  
Combinatorial Interpretations

Some of the classical orthogonal polynomials such as Hermite, Laguerre, Charlier, etc. have been shown to be the generating polynomials for certain combinatorial objects. These combinatorial interpretations are used to prove new identities and generating functions involving these polynomials. In this paper we apply Foata's approach to generating functions for the Hermite polynomials to obtain a triple lacunary generating function. We define renormalized Hermite polynomials $h_n(u)$ by $$\sum_{n= 0}^\infty h_n(u) {z^n\over n!}=e^{uz+{z^2\!/2}}.$$ and give a combinatorial proof of the following generating function: $$ \sum_{n= 0}^\infty h_{3n}(u) {{z^n\over n!}}= {e^{(w-u)(3u-w)/6}\over\sqrt{1-6wz}} \sum_{n= 0}^\infty {{(6n)!\over 2^{3n}(3n)!(1-6wz)^{3n}} {z^{2n}\over(2n)!}}, $$ where $w=(1-\sqrt{1-12uz})/6z=uC(3uz)$ and $C(x)=(1-\sqrt{1-4x})/(2x)$ is the Catalan generating function. We also give an umbral proof of this generating function.

Generating functions and semi-classical orthogonal polynomials

1994 ◽  
Vol 124 (5) ◽  
pp. 1003-1011 ◽  
Author(s):  
Pascal Maroni ◽  
Jeannette Van Iseghem
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Generating Function ◽  
Generating Functions ◽  
Special Form ◽  
Classical Orthogonal Polynomials ◽  
Order Linear ◽  
First Order

An orthogonal family of polynomials is given, and a link is made between the special form of the coefficients of their recurrence relation and a first-order linear homogenous partial differential equation satisfied by the associated generating function. A study is also made of the semiclassical character of such families.

Classical Orthogonal Polynomials as Moments

1997 ◽  
Vol 49 (3) ◽  
pp. 520-542 ◽  
Author(s):  
Mourad E. H. Ismail ◽  
Dennis Stanton
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Functions ◽  
Laguerre Polynomials ◽  
Classical Orthogonal Polynomials ◽  
Ultraspherical Polynomials

AbstractWe show that the Meixner, Pollaczek, Meixner-Pollaczek, the continuous q-ultraspherical polynomials and Al-Salam-Chihara polynomials, in certain normalization, are moments of probability measures.We use this fact to derive bilinear and multilinear generating functions for some of these polynomials. We also comment on the corresponding formulas for the Charlier, Hermite and Laguerre polynomials.

scholarly journals Developments in the Khintchine-Meinardus Probabilistic Method for Asymptotic Enumeration

10.37236/4581 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Boris L. Granovsky ◽  
Dudley Stark
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Probabilistic Method ◽  
Asymptotic Enumeration ◽  
Combinatorial Objects ◽  
Taylor Coefficients ◽  
Gentile Statistics ◽  
Novel Applications

A theorem of Meinardus provides asymptotics of the number of weighted partitions under certain assumptions on associated ordinary and Dirichlet generating functions. The ordinary generating functions are closely related to Euler's generating function $\prod_{k=1}^\infty S(z^k)$ for partitions, where $S(z)=(1-z)^{-1}$. By applying a method due to Khintchine, we extend Meinardus' theorem to find the asymptotics of the Taylor coefficients of generating functions of the form $\prod_{k=1}^\infty S(a_kz^k)^{b_k}$ for sequences $a_k$, $b_k$ and general $S(z)$. We also reformulate the hypotheses of the theorem in terms of the above generating functions. This allows novel applications of the method. In particular, we prove rigorously the asymptotics of Gentile statistics and derive the asymptotics of combinatorial objects with distinct components.

scholarly journals q-Hermite Polynomials and Classical Orthogonal Polynomials

1996 ◽  
Vol 48 (1) ◽  
pp. 43-63 ◽  
Author(s):  
Christian Berg ◽  
Mourad E. H. Ismail
Keyword(s):  
Weight Function ◽  
Generating Functions ◽  
Jacobi Polynomials ◽  
Hermite Polynomials ◽  
Rational Functions ◽  
Classical Orthogonal Polynomials ◽  
Beta Integral ◽  
Orthogonality Relations ◽  
Beta Integrals ◽  
Wilson Polynomials

AbstractWe use generating functions to express orthogonality relations in the form of q-beta. integrals. The integrand of such a q-beta. integral is then used as a weight function for a new set of orthogonal or biorthogonal functions. This method is applied to the continuous q-Hermite polynomials, the Al-Salam-Carlitz polynomials, and the polynomials of Szegö and leads naturally to the Al-Salam-Chihara polynomials then to the Askey-Wilson polynomials, the big q-Jacobi polynomials and the biorthogonal rational functions of Al-Salam and Verma, and some recent biorthogonal functions of Al-Salam and Ismail.

Some properties of the Hermite polynomials

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Higher Order ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Faà Di Bruno Formula ◽  
Derivatives Of

Abstract In this paper, by the Faà di Bruno formula and properties of Bell polynomials of the second kind, the authors reconsider the generating functions of Hermite polynomials and their squares, find an explicit formula for higher-order derivatives of the generating function of Hermite polynomials, and derive explicit formulas and recurrence relations for Hermite polynomials and their squares.

scholarly journals The Borwein Conjecture and Partitions with Prescribed Hook Differences

10.37236/1262 ◽  
1995 ◽  
Vol 3 (2) ◽  
Author(s):  
David A. Bressoud
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Combinatorial Proof

Peter Borwein has conjectured that certain polynomials have non-negative coefficients. In this paper we look at some generalizations of this conjecture and observe how they relate to the study of generating functions for partitions with prescribed hook differences. A combinatorial proof of the generating function for partitions with prescribed hook differences is given.

Jacobi series which converge to zero, with applications to a class of singular partial differential equations

1969 ◽  
Vol 65 (1) ◽  
pp. 101-106 ◽  
Author(s):  
Jet Wimp ◽  
David Colton
Keyword(s):  
Partial Differential Equations ◽  
Weight Function ◽  
Orthogonal Polynomials ◽  
Hermite Polynomials ◽  
Classical Orthogonal Polynomials ◽  
Applied Mathematician ◽  
In Series ◽  
Image Position ◽  
Series Of Functions ◽  
Sequence Of Functions

Expansions in series of functions are one of the most important tools of the applied mathematician, particularly expansions in series of the classical orthogonal polynomials, e.g. Laguerre, Jacobi and Hermite polynomials. In applied problems, the uniqueness of the particular expansion is usually intrinsic to the analysis, and often implicitly assumed. Indeed, in those cases where the functions in the series are orthogonal, uniqueness can often be proved by an argument that runs as follows. Let {φn(x)} (n = 0, 1, 2, …) be a sequence of functions orthogonal with respect to the weight function ρ(x) over the interval [0, 1], and suppose thatthe series being boundedly convergent for 0 ≤ x ≤ 1.

scholarly journals Gegenbauer and Other Planar Orthogonal Polynomials on an Ellipse in the Complex Plane

2020 ◽  
Author(s):  
Gernot Akemann ◽  
Taro Nagao ◽  
Iván Parra ◽  
Graziano Vernizzi
Keyword(s):  
Orthogonal Polynomials ◽  
Complex Plane ◽  
Jacobi Polynomials ◽  
Complete Metric Space ◽  
Hermite Polynomials ◽  
Weighted Bergman Space ◽  
Classical Orthogonal Polynomials ◽  
Space Forms ◽  
Entire Complex ◽  
Fourth Kind

Abstract We show that several families of classical orthogonal polynomials on the real line are also orthogonal on the interior of an ellipse in the complex plane, subject to a weighted planar Lebesgue measure. In particular these include Gegenbauer polynomials $$C_n^{(1+\alpha )}(z)$$ C n ( 1 + α ) ( z ) for $$\alpha >-1$$ α > - 1 containing the Legendre polynomials $$P_n(z)$$ P n ( z ) and the subset $$P_n^{(\alpha +\frac{1}{2},\pm \frac{1}{2})}(z)$$ P n ( α + 1 2 , ± 1 2 ) ( z ) of the Jacobi polynomials. These polynomials provide an orthonormal basis and the corresponding weighted Bergman space forms a complete metric space. This leads to a certain family of Selberg integrals in the complex plane. We recover the known orthogonality of Chebyshev polynomials of the first up to fourth kind. The limit $$\alpha \rightarrow \infty $$ α → ∞ leads back to the known Hermite polynomials orthogonal in the entire complex plane. When the ellipse degenerates to a circle we obtain the weight function and monomials known from the determinantal point process of the ensemble of truncated unitary random matrices.

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