DOCSLIB.ORG

Some Curious Results Involving Certain Polynomials

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi. The Jacobi polynomials are defined via the hypergeometric function as follows: 21 Scholars Research Library Salahuddin and R.K. Khola Euro. J. Appl. Eng. Sci. Res., 201 4, 3 (1):21-30 _____________________________________________________________________________ α β (α + 1) − ( , ) =n −++++α β α 1 z Pzn ()2 Fn 1 (,1 n ;1; ), n! 2 (1.3) α + Where ( 1) n is Pochhammer’s symbol. In this case, the series for the hypergeometric function is finite, therefore one obtains the following equivalent expression: Γ++αn Γ++++− α β (α , β ) = (n 1) n ! ( nmz 1) 1 m Pn ( z ) ∑ ( ) n!(Γ+++α β n 1)= mnm !() − ! Γ++ ( α m 1) 2 m 0 (1.4) Lucas Polynomials The Lucas polynomials are the w-polynomials obtained by setting p(x) = x and q(x) = 1 in the Lucas polyno mials sequence. It is given explicitly by =−n −2 + n +2 + n Lxn ()2[( xx 4 ) + (x x 4 ) ] (1.5 ) The first few are = L1( x ) x L( x )= x 2 + 2 2 (1.6) =3 + Lx3 ( ) x 3 x =4 + 2 + Lx4 () x 4 x 2 Generalized Harmonic Number The generalized harmonic number of order n of m is given by n (m ) = 1 H n ∑ m k=1 k (1.7) In the limit of n → ∞ the generalized harmonic number converges to the Riemann zeta function (m ) = ς limHn ( m ) n→∞ (1.8) 22 Scholars Research Library Salahuddin and R.K. Khola Euro. J. Appl. Eng. Sci. Res., 2014, 3 (1):21-30 _____________________________________________________________________________ Bernoulli Polynomial In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and Hurwitz zeta function. This is in large part because they are an Appell sequence, i.e. a Sheffer sequence for the ordinary derivative operator. Unlike orthogonal polynomials, the Bernoulli polynomials are remarkable in that the number of crossing of the x-axis in the unit interval does not go up as the degree of the polynomials goes up. In the limit of large degree, the Bernoulli polynomials, appropriately scaled, approach the sine and cosine functions. Explicit formula of Bernoulli polynomials is n n! Bx( ) = bx k , for n 0, where b are the Bernoulli numbers. n∑ − n− k ≥ k k =0 k!() n k ! The generating function for the Bernoulli polynomials is text∞ t n = B( x ) t − ∑ n e1n=0 n ! (1.9) Gegenbauer polynomials (ɑ) In Mathematics, Gegenbauer polynomials or ultraspherical polynomials C n (x) are orthogonal polynomials on the α −1 interval [-1,1] with respect to the weight function (1− x2 ) 2 . They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbauer . Explicitly , n 2 α Γ(n − k + α ) − C() () x= (1) − k (2) z ( n 2 k ) n ∑ Γα − k =0 ( )!(k n 2)! k (1.10) 23 Scholars Research Library Salahuddin and R.K. Khola Euro. J. Appl. Eng. Sci. Res., 2014, 3 (1):21-30 _____________________________________________________________________________ Laguerre polynomials The Laguerre polynomials are solutions Ln ( x ) to the Laguerre differential equation xy''+− (1 x ) y ' +λ y = 0 , which is a special case of the more general associated Laguerre differential equation, defined by xy''(++−ν 1 xy )' + λ y = 0 , where λ and ν are real numbers with ν =0. The Laguerre polynomials are given by the sum n − k = ( 1)n ! k Ln ( x ) ∑ x = k! kn !()− k ! k 0 (1.11) Hermite polynomials The Hermite polynomials Hn ( x ) are set of orthogonal polynomials over the domain (-∞,∞) with weighting − 2 function e x . The Hermite polynomials Hn ( x ) can be defined by the contour integral = n! −+r2 2 rz −− n 1 Hzn ( ) ∫ e tdt , 2πi Where the contour incloses the origin and is traversed in a counterclockwise direction 24 Scholars Research Library Salahuddin and R.K. Khola Euro. J. Appl. Eng. Sci. Res., 201 4, 3 (1):21-30 _____________________________________________________________________________ (Arfken 1985, p. 416). The first few Hermite polynomials are = H0 ( x ) 1 = H1( x ) 2 x H() x= 4 x 2 − 2 2 (1.12) =3 − Hx3 () 8 x 12 x =4 − 2 + Hx4 () 16 x 48 x 12 Legendre function of the first kind The Legendre po lynomials, sometimes called Legendre functions of the first kind, Legendre coefficients, or zonal harmonics (Whittaker and Watson 1990, p. 302), are solutions to the Legendre differential equation . If l is an integer, they are polynomials . The Legendre polyn omials Pn ( x ) are illustrated above for and n=1, 2, ..., 5. The Legendre polynomials Pn ( x ) can be defined by the contour integral −1 1 − − = − + 22 n 1 Pzn ( )∫ (1 2 tzt ) tdt , (1.13) 2πi where the contour encloses the origin and i s traversed in a counterclockwise direction (Arfken 1985, p. 416). Legendre function of the second kind 25 Scholars Research Library Salahuddin and R.K. Khola Euro. J. Appl. Eng. Sci. Res., 201 4, 3 (1):21-30 _____________________________________________________________________________ The second solution to the Legendre differential equation . The Leg endre functions of the second kind satisfy the same recurrence relation as the Legendre polynomials . The first few are 1 1+ x Q( x ) = ln( ) 0 2 1− x x 1+ x Q( x ) = ln( )-1 (1 .14) 1 2 1− x 3x2 − 1 1+ x 3x Q( x ) = ln( )- 2 4 1− x 2 Chebyshev polynomial of the first kind The Chebyshev polynomials of the first kind are a set of orthogonal polynomials defined as the solutions to the Chebyshev differential equation and denoted Tn ( x ) . They are used as an approximation to a least squares fit, and are a special case of the Gegenbauer polynomial with . They are also intimately connected with trigonometric multiple-angle formulas . The Chebyshev polynomial of the first kind Tn ( z ) can be defined by the contour integral 1 (1−t2 ) t −n − 1 = , (1.15) Tn ( z ) ∫ 2 dt 4πi 1− 2 tzt + where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416). The first few Chebyshev polynomials of the first kind are = T0 ( x ) 1 = T1( x ) x =2 − T2 () x 2 x 1 =3 − Tx3 () 4 x 3 x A beautiful plot can be obtained by plotting Tn ( x ) radially, increasing the radius for each value of n , and filling in the areas between the curves (Trott 1999, pp. 10 and 84). 26 Scholars Research Library Salahuddin and R.K. Khola Euro. J. Appl. Eng. Sci. Res., 201 4, 3 (1):21-30 _____________________________________________________________________________ The Chebyshev polynomials of the first kind are defined through the identity Tn(cos θ)=cos nθ. Chebyshev polynomial of the second kind A modified set of Chebyshev polynomials defined by a slightly different generating function . They arise in the development of four-dimensional spherical harmonics in angular momentum theory. They are a special case of the Gegenbauer polynomial with . They are also intimately connected with trigonometric multiple-angle formulas. The first few Cheby shev polynomials of the second kind are = U0 ( x ) 1 U( x )= 2 x 1 (1.16) =2 − U2 () x 4 x 1 =3 − Ux3 () 8 x 4 x 27 Scholars Research Library Salahuddin and R.K. Khola Euro. J. Appl. Eng. Sci. Res., 2014, 3 (1):21-30 _____________________________________________________________________________ Euler polynomial The Euler polynomial En ( x ) is given by the Appell sequence with 1 g( t )= ( e t + 1) , 2 giving the generating function 2ext∞ t n ≡ E( x ) . (1.17) t + ∑ n e1n=0 n ! The first few Euler polynomials are = E0 ( x ) 1 1 E( x ) = x − 1 2 Ex( ) = x2 − x 2 (1.18) 3 1 Ex( ) = x3 − x 2 + 3 2 4 1. MAIN RESULTS − 2 1− 2n 2 −−n− − n + xnF(2 1 ( ,;; x ) 4 x 1 − L( x ) 1 + x n dx = 2 n n + C ∫ 1 − 2(n 4) (2.1) − 1− 1n 1 −−n− − n + nx2 F 1 (,;; x ) 2 x x 1 − H(x ) 1+ x n dx = 2 n n + C ∫ 1 − (n 2) (2.2) − + −n 1 −1 − 2n 2 − −n ∫ B1( x ) 1 x dx = x[ (2)nnxF2 1 ( ,; ; x ) 2(n− 4)( n − 2) 2 n n 28 Scholars Research Library Salahuddin and R.K.
Load more

Recommended publications
  • 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.
    [Show full text]
  • New Bell–Sheffer Polynomial Sets
    axioms Article New Bell–Sheffer Polynomial Sets Pierpaolo Natalini 1,* and Paolo Emilio Ricci 2 1 Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Largo San Leonardo Murialdo, 1, 00146 Roma, Italy 2 Sezione di Matematica, International Telematic University UniNettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy; [email protected] * Correspondence: [email protected] Received: 20 July 2018; Accepted: 2 October 2018; Published: 8 October 2018 Abstract: In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers, and several integer sequences related to them, have been studied. The method used in previous articles, and even in the present one, traces back to preceding results by Dattoli and Ben Cheikh on the monomiality principle, showing the possibility to derive explicitly the main properties of Sheffer polynomial families starting from the basic elements of their generating functions. The introduction of iterated exponential and logarithmic functions allows to construct new sets of Bell–Sheffer polynomials which exhibit an iterative character of the obtained shift operators and differential equations. In this context, it is possible, for every integer r, to define polynomials of higher type, which are linked to the higher order Bell-exponential and logarithmic numbers introduced in preceding papers. Connections with integer sequences appearing in Combinatorial analysis are also mentioned. Naturally, the considered technique can also be used in similar frameworks, where the iteration of exponential and logarithmic functions appear. Keywords: Sheffer polynomials; generating functions; monomiality principle; shift operators; combinatorial analysis 1. Introduction In recent articles [1,2], new sets of Sheffer [3] and Brenke [4] polynomials, based on higher order Bell numbers [2,5–7], have been studied.
    [Show full text]
  • New Formulae of Squares of Some Jacobi Polynomials Via Hypergeometric Functions
    Hacettepe Journal of Mathematics and Statistics Volume 46 (2) (2017), 165 176 New formulae of squares of some Jacobi polynomials via hypergeometric functions W.M. Abd- Elhameed∗ y Abstract In this article, a new formula expressing explicitly the squares of Jacobi polynomials of certain parameters in terms of Jacobi polynomials of ar- bitrary parameters is derived. The derived formula is given in terms of ceratin terminating hypergeometric function of the type 4F3(1). In some cases, this 4F3(1) can be reduced by using some well-known re- duction formulae in literature such as Watson's and Pfa-Saalschütz's identities. In some other cases, this 4F3(1) can be reduced by means of symbolic computation, and in particular Zeilberger's, Petkovsek's and van Hoeij's algorithms. Hence, some new squares formulae for Jacobi polynomials of special parameters can be deduced in reduced forms which are free of any hypergeometric functions. Keywords: Jacobi polynomials; linearization coecients; generalized hypergeo- metric functions; computer algebra, standard reduction formulae 2000 AMS Classication: 33F10; 33C20; 33Cxx; 68W30 Received : 23.02.2016 Accepted : 19.05.2016 Doi : 10.15672/HJMS.20164518618 1. Introduction The Jacobi polynomials are of fundamental importance in theoretical and applied mathematical analysis. The class of Jacobi polynomials contains six well-known fami- lies of orthogonal polynomials, they are, ultraspherical, Legendre and the four kinds of Chebyshev polynomials. The Jacobi polynomials in general and their six special polyno- mials in particular are extensively employed in obtaining numerical solutions of ordinary, fractional and partial dierential equations. In this respect, these polynomials are em- ployed for the sake of obtaining spectral solutions for various kinds of dierential equa- tions.
    [Show full text]
  • Some Classical Multiple Orthogonal Polynomials
    Some classical multiple orthogonal polynomials ∗ Walter Van Assche and Els Coussement Department of Mathematics, Katholieke Universiteit Leuven 1 Classical orthogonal polynomials One aspect in the theory of orthogonal polynomials is their study as special functions. Most important orthogonal polynomials can be written as terminating hypergeometric series and during the twentieth century people have been working on a classification of all such hypergeometric orthogonal polynomial and their characterizations. The very classical orthogonal polynomials are those named after Jacobi, Laguerre, and Hermite. In this paper we will always be considering monic polynomials, but in the literature one often uses a different normalization. Jacobi polynomials are (monic) polynomials of degree n which are orthogonal to all lower degree polynomials with respect to the weight function (1 x)α(1+x)β on [ 1, 1], where α,β > 1. The change of variables x 2x 1 gives Jacobi polynomials− on [0, 1]− for the weight function− w(x) = xβ(1 x)α, and we7→ will− denote (α,β) − these (monic) polynomials by Pn (x). They are defined by the orthogonality conditions 1 P (α,β)(x)xβ(1 x)αxk dx =0, k =0, 1,...,n 1. (1.1) n − − Z0 (α) The monic Laguerre polynomials Ln (x) (with α > 1) are orthogonal on [0, ) to all − α x ∞ polynomials of degree less than n with respect to the weight w(x)= x e− and hence satisfy the orthogonality conditions ∞ (α) α x k Ln (x)x e− x dx =0, k =0, 1,...,n 1. (1.2) 0 − arXiv:math/0103131v1 [math.CA] 21 Mar 2001 Z Finally, the (monic) Hermite polynomials Hn(x) are orthogonal to all lower degree polyno- x2 mials with respect to the weight function w(x)= e− on ( , ), so that −∞ ∞ ∞ x2 k H (x)e− x dx =0, k =0, 1,...,n 1.
    [Show full text]
  • 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.
    [Show full text]
  • The Relation of the D-Orthogonal Polynomials to the Appell Polynomials
    View metadata, citation and similar papers at core.ac.uk brought to you byCORE provided by Elsevier - Publisher Connector JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS ELSEVIER Journal of Computational and Applied Mathematics 70 (1996) 279-295 The relation of the d-orthogonal polynomials to the Appell polynomials Khalfa Douak Universitb Pierre et Marie Curie, Laboratoire d'Analyse Numbrique, Tour 55-65, 4 place Jussieu, 75252 Paris Cedex 05, France Received 10 November 1994; revised 28 June 1995 Abstract We are dealing with the concept of d-dimensional orthogonal (abbreviated d-orthogonal) polynomials, that is to say polynomials verifying one standard recurrence relation of order d + 1. Among the d-orthogonal polynomials one singles out the natural generalizations of certain classical orthogonal polynomials. In particular, we are concerned, in the present paper, with the solution of the following problem (P): Find all polynomial sequences which are at the same time Appell polynomials and d-orthogonal. The resulting polynomials are a natural extension of the Hermite polynomials. A sequence of these polynomials is obtained. All the elements of its (d + 1)-order recurrence are explicitly determined. A generating function, a (d + 1)-order differential equation satisfied by each polynomial and a characterization of this sequence through a vectorial functional equation are also given. Among such polynomials one singles out the d-symmetrical ones (Definition 1.7) which are the d-orthogonal polynomials analogous to the Hermite classical ones. When d = 1 (ordinary orthogonality), we meet again the classical orthogonal polynomials of Hermite. Keywords: Appell polynomials; Hermite polynomials; Orthogonal polynomials; Generating functions; Differential equations; Recurrence relations AMS classification: 42C99; 42C05; 33C45 O.
    [Show full text]
  • 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.
    [Show full text]
  • Q-Special Functions, a Tutorial
    q-Special functions, a tutorial TOM H. KOORNWINDER Abstract A tutorial introduction is given to q-special functions and to q-analogues of the classical orthogonal polynomials, up to the level of Askey-Wilson polynomials. 0. Introduction It is the purpose of this paper to give a tutorial introduction to q-hypergeometric functions and to orthogonal polynomials expressible in terms of such functions. An earlier version of this paper was written for an intensive course on special functions aimed at Dutch graduate students, it was elaborated during seminar lectures at Delft University of Technology, and later it was part of the lecture notes of my course on “Quantum groups and q-special functions” at the European School of Group Theory 1993, Trento, Italy. I now describe the various sections in some more detail. Section 1 gives an introduction to q-hypergeometric functions. The more elementary q-special functions like q-exponential and q-binomial series are treated in a rather self-contained way, but for the higher q- hypergeometric functions some identities are given without proof. The reader is referred, for instance, to the encyclopedic treatise by Gasper & Rahman [15]. Hopefully, this section succeeds to give the reader some feeling for the subject and some impression of general techniques and ideas. Section 2 gives an overview of the classical orthogonal polynomials, where “classical” now means “up to the level of Askey-Wilson polynomials” [8]. The section starts with the “very classical” situation of Jacobi, Laguerre and Hermite polynomials and next discusses the Askey tableau of classical orthogonal polynomials (still for q = 1).
    [Show full text]
  • Pp. 101–130. the Classical Umbral Calculus
    Lecture Notes of Seminario Interdisciplinare di Matematica Vol. 8(2009), pp. 101–130. The classical umbral calculus: She↵er sequences Elvira Di Nardo, Heinrich Niederhausen and Domenico Senato Abstract1. Following the approach of Rota and Taylor [17], we present an innovative theory of She↵er sequences in which the main properties are encoded by using umbrae. This syntax allows us noteworthy computational simplifications and conceptual clarifica- tions in many results involving She↵er sequences. To give an indication of the e↵ectiveness of the theory, we describe applications to the well-known connection constants problem, to Lagrange inversion formula and to solving some recurrence relations. 1. Introduction As well known, many polynomial sequences like Laguerre polynomials, first and second kind Meixner polynomials, Poisson-Charlier polynomials and Stirling poly- nomials are She↵er sequences. She↵er sequences can be considered the core of umbral calculus: a set of tricks extensively used by mathematicians at the begin- ning of the twentieth century. Umbral calculus was formalized in the language of the linear operators by Gian- Carlo Rota in a series of papers (see [15], [16], and [14]) that have produced a plenty of applications (see [1]). In 1994 Rota and Taylor [17] came back to foundation of umbral calculus with the aim to restore, in light formal setting, the computational power of the original tools, heuristical applied by founders Blissard, Cayley and Sylvester. In this new setting, to which we refer as the classical umbral calculus, there are two basic devices. The first one is to represent a unital sequence of num- bers by a symbol ↵, called an umbra, that is, to represent the sequence 1,a1,a2,..
    [Show full text]
  • Orthogonal Polynomials and Classical Orthogonal Polynomials
    International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue 10, October 2018, pp. 1613–1630, Article ID: IJMET_09_10_164 Available online at http://iaeme.com/Home/issue/IJMET?Volume=9&Issue=10 ISSN Print: 0976-6340 and ISSN Online: 0976-6359 © IAEME Publication Scopus Indexed ORTHOGONAL POLYNOMIALS AND CLASSICAL ORTHOGONAL POLYNOMIALS DUNIA ALAWAI JARWAN Education for Girls College, Al-Anbar University, Ministry of Higher Education and Scientific Research, Iraq ABSTRACT The focus of this project is to clarify the concept of orthogonal polynomials in the case of continuous internal and discrete points on R and the Gram – Schmidt orthogonalization process of conversion to many orthogonal limits and the characteristics of this method. We have highlighted the classical orthogonal polynomials as an example of orthogonal polynomials because of they are great importance in physical practical applications. In this project, we present 3 types (Hermite – Laguerre – Jacobi) of classical orthogonal polynomials by clarifying the different formulas of each type and how to reach some formulas, especially the form of the orthogonality relation of each. Keywords: Polynomials, Classical Orthogonal, Monic Polynomial, Gram – Schmidt Cite this Article Dunia Alawai Jarwan, Orthogonal Polynomials and Classical Orthogonal Polynomials, International Journal of Mechanical Engineering and Technology, 9(10), 2018, pp. 1613–1630. http://iaeme.com/Home/issue/IJMET?Volume=9&Issue=10 1. INTRODUCTION The mathematics is the branch where the lots of concepts are included. An orthogonality is the one of the concept among them. Here we focuse on the orthogonal polynomial sequence. The orthogonal polynomial are divided in two classes i.e. classical orthogonal polynomials, Discrete orthogonal polynomials and Sieved orthogonal polynomials .There are different types of classical orthogonal polynomials such that Jacobi polynomials, Associated Laguerre polynomials and Hermite polynomials.
    [Show full text]
  • Rodrigues Formula for Jacobi Polynomials on the Unit Circle
    RODRIGUES FORMULA FOR JACOBI POLYNOMIALS ON THE UNIT CIRCLE MASTERS THESIS Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of the Ohio State University By Griffin Alexander Reiner-Roth Graduate Program in Mathematics The Ohio State University 2013 Master's Examination Committe: Professor Rodica Costin, Advisor Professor Barbara Keyfitz c Copyright by Griffin Alexander Reiner-Roth 2013 ABSTRACT We begin by discussing properties of orthogonal polynomials on a Borel measur- able subset of C. Then we focus on Jacobi polynomials and give a formula (analogous to that of [5]) for finding Jacobi polynomials on the unit circle. Finally, we consider some examples of Jacobi polynomials and demonstrate different methods of discov- ering them. ii ACKNOWLEDGMENTS I would like to thank my advisor, Dr. Rodica Costin, for all her help on this thesis; for finding a really interesting and doable project for me, for spending so much time explaining the basics of orthogonal polynomials to me, and for perfecting this paper. I also thank my parents for making sure I never starved. iii VITA 2011 . B.A. Mathematics, Vassar College 2011-Present . Graduate Teaching Associate, Department of Mathematics, The Ohio State Univer- sity FIELDS OF STUDY Major Field: Mathematics iv TABLE OF CONTENTS Abstract . ii Acknowledgments . iii Vita......................................... iv List of Tables . vii CHAPTER PAGE 1 Introduction . .1 1.1 Historical Background . .1 1.2 Orthogonality in Hilbert Spaces . .4 1.3 Orthogonal Polynomials with Respect to Measures . .5 1.4 Approximation by Orthogonal Polynomials . 12 1.5 Classical Orthogonal Polynomials . 15 2 Jacobi Polynomials .
    [Show full text]
  • Hermite Polynomials 1 Hermite Polynomials
    Hermite polynomials 1 Hermite polynomials In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical analysis as Gaussian quadrature; in finite element methods as shape functions for beams; and in physics, where they give rise to the eigenstates of the quantum harmonic oscillator. They are also used in systems theory in connection with nonlinear operations on Gaussian noise. They were defined by Laplace (1810) [1] though in scarcely recognizable form, and studied in detail by Chebyshev (1859).[2] Chebyshev's work was overlooked and they were named later after Charles Hermite who wrote on the polynomials in 1864 describing them as new.[3] They were consequently not new although in later 1865 papers Hermite was the first to define the multidimensional polynomials. Definition There are two different ways of standardizing the Hermite polynomials: • The "probabilists' Hermite polynomials" are given by , • while the "physicists' Hermite polynomials" are given by . These two definitions are not exactly identical; each one is a rescaling of the other, These are Hermite polynomial sequences of different variances; see the material on variances below. The notation He and H is that used in the standard references Tom H. Koornwinder, Roderick S. C. Wong, and Roelof Koekoek et al. (2010) and Abramowitz & Stegun. The polynomials He are sometimes denoted by H , n n especially in probability theory, because is the probability density function for the normal distribution with expected value 0 and standard deviation 1.
    [Show full text]