DOCSLIB.ORG

(X) 1. Introduction Orthogonal Families of Polynomial Are Impo

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
(X) 1. Introduction Orthogonal Families of Polynomial Are Impo Orthogonality for a class of generalised Jacobi polynomial $P^{\alpha,\beta}_{\nu}(x)$ Item Type Article Authors Ford, Neville J.; Moayyed, H.; Rodrigues, M. M. Citation Ford, N. J., Moayyed, H. & Rodrigues, M. M. (2018) Orthogonality for a class of generalised Jacobi polynomial $P^{\alpha, \beta}_{\nu}(x)$. Fractional Differential Calculus, 8(1), 95-110 DOI 10.7153/fdc-2018-08-06 Publisher Ele-Math Journal Fractional Differential Calculus Download date 02/10/2021 17:29:02 Item License http://creativecommons.org/licenses/by-nc-nd/4.0/ Link to Item http://hdl.handle.net/10034/620536 F ractional D ifferential C alculus www.ele-math.com ORTHOGONALITY FOR A CLASS OF a;b GENERALISED JACOBI POLYNOMIAL Pn (x) N. J. FORD, H. MOAYYED AND M. M. RODRIGUES Submitted to Fracttional Differ. Calc. Abstract. This work considers g-Jacobi polynomials, a fractional generalisation of the classical Jacobi polynomials. We discuss the polynomials and compare some of their properties to the classical case. The main result of the paper is to show that one can derive an orthogonality prop- (a;b) erty for a sub-class of g-Jacobi polynomials Pn . The paper concludes with an application in modelling of ophthalmic surfaces. 1. Introduction Orthogonal families of polynomial are important both from theoretical and practi- cal viewpoints. They are widely used when approximating functions from mathematics and other sciences and they allow us to solve and interpret solutions of certain differen- tial equations. (a;b) The Jacobi Polynomials Pn are a classical family of orthogonal polynomials (see [7], [23]) that has been used in many applications due to its ability to approximate general classes of function. The Gegenbauer polynomials (which are special cases of Jacobi polynomials) are orthogonal polynomials on the interval [−1;1] with respect to the weight function (1 − x2)a−1=2 . They are a generalisation of the Legendre and Chebyshev polynomi- l als. Further, the Gegenbauer polynomials fCn (t)g constitute a complete orthogonal − 1 system of the real (or complex) L2−space L2 [−1;1];(1 − x2)n 2 dx with respect to the (1 − x2)a−1=2 on the interval [−1;1] (see [24]). One can also consider the question of orthogonality over a semi-infinite interval. The authors of [6] discuss this property for the Jacobi polynomials and this was also considered by V. Romanovski [20] and (more recently) by S.D. Bajpai[4]. In[11] the authors introduced a family of generalised Jacobi polynomials/functions with indexes a;b 2 R which are mutually orthogonal with respect to the corresponding Mathematics subject classification (2010): 33F05, 33C90, 12E10. Keywords and phrases: Jacobi polynomials; Approximation; Optic modelling.. The work of second and third authors were supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT– Fundac¸ao˜ para a Cienciaˆ e a Tecnologia”), within project UID/MAT/ 0416/2013. Additionally the research grant (BIP- LtPICS-2016A) of the second author was also supported by FCT / MEC through national funds and where appropriate co-financed by the FEDER, under the PT2020 Partnership Agreement.. 1 Jacobi weights and which inherit selected important properties of the classical Jacobi polynomials. a;b;M;N ¥ In [13] the authors deal with generalised Jacobi polynomials fPn (x)gn=0 , a;b but the generalisation we consider in this paper, to the form fPn (x)g with n non in- teger is much more recent (see [15]). We will show that these fractional generalisations can have important properties of use in applications. The classical Jacobi Polynomials are usually defined by a Rodrigues’ type formula dn h i P(a;b)(x) = (−2)n(n!)−1(1 − x)−a (1 + x)−b (1 − x)n+a (1 + x)n+b ; n dxn where a > −1;b > −1 (see [7]). In [15] the authors defined the generalised Jacobi (or g-Jacobi) functions as an extension of the Jacobi polynomials. In the previous formula they replaced the integer derivative by the Riemann-Liouville fractional of order n 2 R. They also derived some formal properties of this new family of polynomials. The main theme of this paper is to find a sub-class of the g-Jacobi Polynomials, (a;b) Pn , with n non-integer, where a orthogonality relation holds, and to show (through an application in optics) how this new class of orthogonal polynomials can have useful practical applications in function aproximation. The paper is organised as follows. In the second section, various definitions and properties are recalled. In the third section, we prove the orthogonality property for a certain class of generalised Jacobi polynomials and some of their properties will be presented in section four. Finally, we compare the behaviour of Zernike, classical and modified Jacobi polynomials in modelling of ophthalmic surfaces. 2. Preliminaries 2.1. Special functions Here we recall some aspects of some of the special functions that feature in this paper: the Gamma function and the Pochhammer symbol. We recall that the Gamma function G(s) is defined by the integral (see [3]) Z ¥ G(s) = e−tts−1dt: (1) 0 The integral (1) defines a holomorphic function in the half-plane Re(s) > 0. In addition G satisfies the functional equation G(s + 1) = sG(s); Re(s) > 0: (2) Hence, since G(1) = 1, we have G(n + 1) = n! for all n 2 N. The reflection formula for the Gamma function establishes a relation between the sine and gamma functions, i.e., p csc(pa) G(−a) = − ; a 2= : G(a + 1) Z 2 The Pochhammer symbol is defined as see [3] G(a + k) (a) := ; k G(a) − for all a 2 CnZ0 . The regularised hypergeometric function 2F˜1(a;b;c;z) is defined by means of a hypergeometric series by the expression ¥ a b ˜ ( )k ( )k k 2F1(a;b;c;z) = ∑ z ; jzj ≤ 1: (3) k=0 G(c + k) k! It is related to the hypergeometric function 2F1 by the formula 1 F˜ (a;b;c;z) = F (a;b;c;z); (4) 2 1 G(c) 2 1 and the hypergeometric function 2F1 has the following series representation ¥ (a)k (b)k k 2F1(a;b;c;z) = ∑ z ; jzj ≤ 1: (5) k=0 (c)k k! When z = 1 the hypergeometric function 2F1 assumes the particular form (see formula (7.3.5.2) in [18]) G(c) G(c − a − b) F (a;b;c;1) = : (6) 2 1 G(c − a) G(c − b) 2.2. Generalised Jacobi polynomials We start by reviewing and defining the Jacobi polynomials. (a;b) We recall that the classical Jacobi polynomials Pn (x) (n ≥ 0) are defined by dn h i P(a;b)(x) = (−2)n(n!)−1(1 − x)−a (1 + x)−b (1 − x)n+a (1 + x)n+b : (7) n dxn a b Let wa;b (x) := (1 − x) (1 + x) be the Jacobi weight function. For a;b > −1 the Jacobi polynomials are mutually orthogonal in L2 ([−1;1]), i.e., wa;b 1 a+b+1 Z (a;b) (a;b) 2 G(a + n + 1)G(b + n + 1) Pm (x)Pn (x)wa;b (x)dx = dmn ; (8) −1 (2n + a + b + 1)G(n + a + b + 1)n! where dmn denotes the Kronecker delta m;n 2 N0 . This class of Jacobi weight func- tions leads to Jacobi polynomials with many attractive properties that are not shared by general orthogonal polynomials. The generalised Jacobi polynomials may be considered as fractional index gener- alisations of the classical Jacobi polynomials. These polynomials appear as a special case of fractional Gauss functions, defined as solutions of the fractional generalisation of the Gauss hypergeometric equation (see [10]). The representation of the generalised Jacobi polynomial in terms of the hypergeo- metric function 2F1 is given by 3 THEOREM 1. (c.f. [15]) The g-Jacobi Polynomial can be represented by G(a + n + 1) (1 − x) Pa;b (x) = F −n;a + b + n + 1;a + 1; : (9) n G(n + 1))G(1 + a) 2 1 2 Moreover, (9) satisfies the following differential equation (1 − x2)Y 00(x) + (b − a − (a + b + 2)x)Y 0(x) + n(n + 2l)Y(x) = 0: (10) The g-Jacobi polynomials were introduced in [15]. There, the authors replaced the integer derivatives in (7) by the Riemann-Liouville fractional derivative Dn , obtaining (−1)n h i Pa;b (x) = Dn (1 − x)n+a (1 + x)n+b ; (11) n G(n + 1)2n (1 − x)a (1 + x)b where the Riemann-Liouville fractional derivative of order a is defined as 8 n 1 d R t f (t) dt; n − 1 < a < n 2 <> G(n−a) dtn a (t−t)a+1−n N Da f (t) = (12) :> dn dtn ; a = n 2 N (see [21]). In [15] the authors presented the following explicit formula for the g-Jacobi poly- nomials THEOREM 2. (cf. [15]) For the generalised Jacobi functions the following repre- sentation holds ¥ ( ; ) n + b n + a P a b (x) = 2−n (x + 1)n−k(x − 1)k (13) n ∑ k n − k k=0 with n > 0;a > −1;b > −1 and where a G(1 + a) = b G(1 + b)G(1 + a − b) is the binomial coefficient with real arguments. We know that the classical Jacobi polynomials satisfy an orthogonality property. In the following example we can see that the orthogonality property (in general) for the g-Jacobi polynomials is not applicable.
Load more

Recommended publications
  • 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]
  • 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]
  • 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]
  • Moments of Classical Orthogonal Polynomials
    Moments of Classical Orthogonal Polynomials zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr.rer.nat) im Fachbereich Mathematik der Universität Kassel By Patrick Njionou Sadjang ????? Ph.D thesis co-supervised by: Prof. Dr. Wolfram Koepf University of Kassel, Germany and Prof. Dr. Mama Foupouagnigni University of Yaounde I, Cameroon October 2013 Tag der mündlichen Prüfung 21. Oktober 2013 Erstgutachter Prof. Dr. Wolfram Koepf Universität Kassel Zweitgutachter Prof. Dr. Mama Foupouagnigni University of Yaounde I Abstract The aim of this work is to find simple formulas for the moments mn for all families of classical orthogonal polynomials listed in the book by Koekoek, Lesky and Swarttouw [30]. The generating functions or exponential generating functions for those moments are given. To my dear parents Acknowledgments Foremost, I would like to express my sincere gratitude to my advisors Prof. Dr. Wolfram Koepf and Prof. Dr. Mama Foupouagnigni for the continuous support of my Ph.D study and research, for their patience, motivation, enthusiasm, and immense knowledge. Their guidance helped me in all the time of research and writing of this thesis. I could not have imagined having better advisors and mentors for my Ph.D study. I am grateful to Prof. Dr. Mama Foupouagnigni for enlightening me the first glance of re- search. My sincere thanks also go to Prof. Dr. Wolfram Koepf for offering me the opportunity to visit the University of Kassel where part of this work has been written. I acknowledge the financial supports of the DAAD via the STIBET fellowship which en- abled me to visit the Institute of Mathematics of the University of Kassel.
    [Show full text]
  • Jacobi Polynomials Via Pictures
    Jacobi Polynomials via Pictures Yao-Rui Notes for Pizza Seminar Abstract Jacobi polynomials are an important class of orthogonal polynomials that includes examples like the Chebyshev and Legendre polynomials. The purpose of this expository note is to flesh out two pictorial appearances of these polynomials in physics and combinatorial number theory. We give a quick overview of Jacobi polynomials, followed by applications in electrostatics and dessins d'enfants. 1 Some Facets of Jacobi Polynomials In this section we will introduce the bare minimum on Jacobi polynomials needed to understand this talk. A good reference on this is [4]. 1.1 Chebyshev Polynomials Let start with an example of Jacobi polynomials. Recall from high school calculus that one can expand cos(nθ) as a polynomial in cos(θ). For example, cos(2θ) = 2 cos2(θ) − 1; cos(3θ) = 4 cos3(θ) − 3 cos(θ); and more generally, using Euler's formula on complex numbers, bn=2c X n cos(nθ) = (cos2(θ) − 1)j cosn−2j(θ): 2j j=0 By replacing cos(θ) with a variable, this gives us the Chebyshev polynomials: bn=2c X n T (x) = (x2 − 1)jxn−2j: n 2j j=0 (There is a completely analogous function by expanding sin(nθ); we will omit this here.) The Chebyshev polynomials have a number of properties, and the first two will be essential later. • The zeroes of Tn are simple and lie in (−1; 1). In this case we can explicitly write them down: (1 + 2k)π cos ; k = 0; : : : ; n − 1: 2n Therefore, the non-endpoint critical points of Tn interlaces with the zeros and lies in (−1; 1).
    [Show full text]
  • A New Property of a Class of Jacobi Polynomials
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9939(XX)0000-0 A NEW PROPERTY OF A CLASS OF JACOBI POLYNOMIALS GEORGE CSORDAS, MARIOS CHARALAMBIDES, AND FABIAN WALEFFE Abstract. Polynomials whose coefficients are successive derivatives of a class of Jacobi polynomials evaluated at x = 1 are stable. This yields a novel and short proof of the known result that the Bessel polynomials are stable polyno- mials. Stability preserving linear operators are discussed. The paper concludes with three open problems involving the distribution of zeros of polynomials. 1. Introduction The new property referred to in the title was observed and conjectured (see Conjecture 1 below) while developing a numerical solution for the Navier-Stokes equations [23]. It is related to the fundamental problem of constructing discretiza- tion schemes of continuous problems (involving, for example, boundary value prob- lems for various systems of partial differential equations) in such a manner that the associated eigenvalue problems are free of “spurious eigenvalues”; that is, the eigenvalues are all negative. For the purpose of this paper, it is sufficient to consider the problem of constructing polynomial approximations to the eigenvalue problem d2u/dx2 = λu for 1 < x < 1 with u( 1) = 0. It is well-known that the solutions to this problem consist− of negative eigenvalues± λ with trigonometric eigenfunctions u(x). If un(x) is a polynomial approximation of degree n to u(x), then the residual d2 (1) R (x) := λu (x) u (x) n n − dx2 n is also a polynomial of degree n in x and this relationship can be inverted to obtain [n/2] d2k (2) u (x) = µk+1 R (x) n dx2k n kX=0 where µ = 1/λ and [n/2] denotes the greatest integer less than or equal to n/2.
    [Show full text]
  • [Math.CA] 9 Oct 1995
    UPWARD EXTENSION OF THE JACOBI MATRIX FOR ORTHOGONAL POLYNOMIALS Andre´ Ronveaux and Walter Van Assche Facult´es Universitaires Notre Dame de la Paix, Namur Katholieke Universiteit Leuven March 3, 1995 Abstract. Orthogonal polynomials on the real line always satisfy a three-term re- currence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We investigate new orthogonal polynomials by adding to the Jacobi matrix r new rows and columns, so that the original Jacobi matrix is shifted downward. The r new rows and columns contain 2r new parameters and the newly obtained orthogonal polynomials thus correspond to an upward extension of the Jacobi matrix. We give an explicit expression of the new orthogonal polynomials in terms of the original orthogonal polynomials, their associated polynomials and the 2r new parameters, and we give a fourth order differential equation for these new polynomials when the original orthogonal polynomials are classical. Furthermore we show how the orthogo- nalizing measure for these new orthogonal polynomials can be obtained and work out the details for a one-parameter family of Jacobi polynomials for which the associated polynomials are again Jacobi polynomials. 1. Introduction The construction of families of orthogonal polynomials on the real line from a given system of orthogonal polynomials (or from a given orthogonalizing weight µ) arXiv:math/9510213v1 [math.CA] 9 Oct 1995 has been the subject of various investigations [6], [4], [14], [11], [3], [15], [7], [9], [16], [25]. Let Pn (n = 0, 1, 2,...) be a sequence of monic orthogonal polynomials on the real line, with orthogonality measure µ, then these polynomials satisfy a three-term recurrence relation 2 (1.1) P (x)=(x b )P (x) a P − (x), n 0, n+1 − n n − n n 1 ≥ R 2 with bn and an > 0 and initial conditions P0 = 1, P−1 = 0.
    [Show full text]
  • How Can Mathematicians and Mathematical Historians Help Each Other?
    ---------Richard Askey--------- How Can Mathematicians and Mathematical Historians Help Each Other? 1. Introduction This paper could have a slightly different title with the word How dropped, but I could argue both sides of that question. The present title presumes the optimistic answer, and while we all hope that this is the cor­ rect answer, the present time may not be the right one for this answer. The history of mathematics is not an easy field, and it takes a rare per­ son to be good at it. Keynes supposedly said it took a rare person to be a great economist: one must be a second-rate historian, mathematician, and philosopher. For him second-rate was very good, but not great. The same is probably true about the history of mathematics, except one may not have to be a philosopher. Some great mathematicians have made im­ portant contributions to the history of mathematics, but very few have spent enough time and thought on the history of mathematics to be able to write a first-rate historical account of part of mathematics. One recent exception is A. Weil, who has added an excellent historical account of number theory before Gauss (Weil 1984) to the historical notes he wrote for the Bourbaki volumes (Bourbaki 1974). His paper from the Helsinki Congress (Weil 1980) should also be read by anyone interested in the history of mathematics. Since my training is in mathematics, and my reading of history has been almost random, I have found it useful to think about what history is and how the history of mathematics differs from cultural or political history.
    [Show full text]
  • Shifted Jacobi Polynomials and Delannoy Numbers 3
    SHIFTED JACOBI POLYNOMIALS AND DELANNOY NUMBERS GABOR´ HETYEI A` la m´emoire de Pierre Leroux Abstract. We express a weighted generalization of the Delannoy numbers in terms of shifted Jacobi polynomials. A specialization of our formulas extends a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago, to all Delannoy numbers and certain Jacobi polynomials. Another specialization provides a weighted lattice path enumeration model for shifted Jacobi polynomials, we use this to present a new combinatorial proof of the orthogonality of Jacobi polynomials with natural number parameters. The proof relates the orthogonality of these polynomials to the orthogonality of (generalized) Laguerre polynomials, as they arise in the theory of rook polynomials. We provide a combinatorial proof for the orthogonality of certain Romanovski-Jacobi polynomials with zero first parameter and negative integer second parameter, considered as an initial segment of the list of similarly transformed Jacobi polynomials with the same parameters. We observe that for an odd second parameter one more polynomial may be added to this finite orthogonal polynomial sequence than what was predicted by a classical result of Askey and Romanovski. The remaining transformed Jacobi polynomials in the sequence are either equal to the already listed Romanovski- Jacobi polynomials or monomial multiples of similarly transformed Jacobi polynomials with a positive second parameter. We provide expressions for an analogous weighted generalization of the Schr¨oder numbers in terms of the Jacobi polynomials, and use this model, together with a result of Mansour and Sun, to express the Narayana polynomials in terms of shifted Jacobi polynomials.
    [Show 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.
    [Show full text]