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Moments of Classical Orthogonal Polynomials

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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. I also acknowl- edge the financial support from the Alexander von Humboldt Foundation via the Research- Group Linkage Programme 2009-2012 between the University of Kassel (Germany) and the University of Yaounde I (Cameroon). I thank my colleagues of the Alexander von Humboldt Laboratory of Computational and Educational Mathematics (University of Yaounde I, Cameroon): Daniel D. Tcheutia, Mau- rice Kenfack, Salifou Mboutngam, for the stimulating discussions, and for all the fun we have had in the last four years. Thanks also to my former professors of the University of Yaounde I and the Higher Teach- ers’ Training College of Yaounde, particularly Prof. Dr. Nicolas Gabriel Andjiga, Prof. Dr. Gabriel Nguetseng, Prof. Dr. Norbert Noutchegueme, Prof. Dr. Mama Foupouagnigni, Prof. Dr. Bertrand Tchantcho. My sincere thanks go to Dr. Etienne Le Grand Nana Chiadjeu, for having taken very good care of me during my stay in Germany in 2012. I would also like to thank my family: my parents Sadjang and Noupa Claudine, for giving birth to me at the first place and supporting me spiritually throughout my life. Last but not least, my thanks go to all those who directly or indirectly contributed in any form whatsoever to the realization of this work. Contents Abstract ii Acknowledgments iv 1 Introduction 1 2 Definitions and Miscellaneous Relations 6 2.1 Special functions . 6 2.1.1 Gamma and Beta functions . 6 2.1.2 Hypergeometric functions . 6 2.1.3 Basic hypergeometric functions . 7 2.1.4 q-Exponential functions . 8 2.2 Difference operators . 8 2.2.1 The operators D and r ........................... 8 2.2.2 The operator Dq ................................ 9 2.2.3 The operators D and S ........................... 9 2.2.4 The operators D and S ........................... 10 2.2.5 The operator D# ................................ 10 2.2.6 The operators Dx and Sx .......................... 10 2.3 q-integration . 10 2.3.1 The q-integration in the interval (0; a), a > 0 . 10 2.3.2 The q-integration in the interval (a; 0), a < 0 . 11 2.3.3 The q-integration in the interval (a; ¥), a > 0 . 11 2.3.4 The q-integration in the interval (−¥; a), a < 0 . 11 2.4 Orthogonal polynomials . 12 2.4.1 Classical continuous orthogonal polynomials . 13 2.4.2 Classical discrete orthogonal polynomials . 15 2.4.3 Classical q-discrete orthogonal polynomials . 16 2.4.4 Classical orthogonal polynomials on a quadratic lattice . 18 2.4.5 Classical orthogonal polynomials on a q-quadratic lattice . 19 2.5 Generating functions . 21 3 Moments of Orthogonal Polynomials: Easy Cases 23 3.1 Classical continuous orthogonal polynomials . 23 3.1.1 Jacobi polynomials . 23 3.1.2 Laguerre polynomials . 26 3.1.3 Bessel polynomials . 27 3.1.4 Hermite polynomials . 28 3.2 Classical q-discrete orthogonal polynomials . 29 3.2.1 Little q-Jacobi polynomials . 29 3.2.2 Little q-Legendre polynomials . 30 3.2.3 q-Krawtchouk polynomials . 30 3.2.4 Little q-Laguerre (Wall) polynomials . 31 3.2.5 q-Laguerre polynomials . 32 3.2.6 q-Bessel polynomials . 33 3.2.7 q-Charlier polynomials . 33 Contents vi 4 Inversion Formulas 35 4.1 The methods . 35 4.1.1 The algorithmic method . 35 4.1.2 Inversion results from Verma’s bibasic formula . 36 4.2 Explicit representations of the inversion coefficients . 37 4.2.1 The classical continuous case . 37 4.2.2 The classical discrete case . 38 4.2.3 The classical q-discrete case . 38 4.2.4 The classical quadratic case . 41 4.2.5 The classical q-quadratic case . 42 5 Moments of Orthogonal Polynomials: Complicated Cases 45 5.1 Introduction . 45 5.2 Inversion formula and moments of orthogonal polynomials . 45 5.3 Some connection formulas between some bases . 46 5.3.1 Elementary symmetric polynomials . 46 5.3.2 Connection between xn and xn ....................... 47 n 5.3.3 Connection between x and (x; q)n ..................... 48 n n 5.3.4 Connection formula between x and (x 1)q . 49 n 5.3.5 Connections between (x y)q and (x; q)n . 50 2 n 5.3.6 Connection between (x ) and (a − ix)n(a + ix)n . 50 n 5.3.7 Connection between (x(x + #)) and (−x)n(x + #)n. 51 n 5.3.8 Connection between x and (a + ix)n ................... 52 n iq −iq 5.3.9 Connection between cos q and (ae , ae ; q)n. 54 5.4 Moments and generating functions . 55 5.4.1 The continuous case . 55 5.4.2 The discrete case . 57 5.4.3 The q-discrete case . 61 5.4.4 The quadratic case . 75 5.4.5 The q-quadratic case . 80 Conclusion and Perspectives 95 Chapter 1 Introduction The xyz-axes of three-dimensional space are pairwise orthogonal with each other. This is very convenient since for that reason many formulas are extremely simple. Every point of three-dimensional space is written as linear combination of such orthogonal coordinates. In a similar fashion, many functions can be written as linear combinations of orthogonal polynomials which play the role of the coordinates. For this reason orthogonal polynomials play a very prominent role in applications. Monic polynomial families orthogonal with respect to the measure da(x) 8 Z b <1 if n = m Pn(x)Pm(x)da(x) = kn dmn, kn 6= 0, dmn = ; n, m 2 N, a :0 if n 6= m Z b n are given explicitly in terms of the moments mn = x da(x), n ≥ 0, by [49] a m0 m1 ··· mn m1 m2 ··· mn+1 1 . Pn(x) = . , dn−1 mn−1 mn+1 ··· m2n−1 1 x ··· xn where m0 m1 ··· mn m1 m2 ··· mn+1 . dn = . 6= 0, n ≥ 0. mn−1 mn ··· m2n−1 mn mn+1 ··· m2n The previous representation shows that the moments characterize fully the orthogonal fam- ily (Pn)n. Also, the moments are involved in the representation of the Stieltjes series ¥ m ( ) = n S x ∑ n+1 , n=0 x which is useful for the characterization of families of orthogonal polynomials via the Riccati equation and also for the determination of the measure da(x) by means of the Stieltjes 2 inverse formula [48] 1 Z t a(t) − a(s) = − lim Im(S(x + i y) dx. p y!0+ s For the moments (mn)n2N of an orthogonal family, the generating function is defined by ¥ n G0(z) = ∑ mnz , n=0 while the exponential generating function is defined by ¥ zn G1(z) = ∑ mn , n=0 n! and the q-exponential generating functions are defined by ¥ zn G (z) = m , 2 ∑ n ( ) n=0 q; q n ¥ zn G (z) = m . 3 ∑ n [ ] n=0 n q! Generating functions, exponential generating functions and q-exponential generating func- tions contain the information of all moments of the orthogonal polynomial family at the same time. Despite the important role that the moments play in various topics of orthogonal poly- nomials and applications to other domains such as statistics and probability theory, no ex- haustive repository of moments for the well-known classical orthogonal polynomials can be found in the literature. The book by Koekoek, Lesky and Swarttouw [30] which is one of the best and most famous documents containing almost all kinds of formulas and relations for various classical orthogonal polynomials does not provide information about the mo- ments. It becomes therefore imperative to investigate this topic in order to complete such missing important information. Classical orthogonal polynomials of a continuous, discrete and q-discrete variable are known to be orthogonal with respect to a weight function r satisfying respectively the Pearson, the discrete Pearson and the q-discrete Pearson equation 0 (s(x)r(x)) = t(x)r(x), (1.1) D (s(x)r(x)) = t(x)r(x), (1.2) Dq (s(x)r(x)) = t(x)r(x), (1.3) where s(x) = ax2 + bx + c is a non-zero polynomial of degree at most two, t(x) = dx + e ( ) = f (qx)− f (x) 6= is a first degree polynomial, Dq is the Hahn operator Dq f x (q−1)x , q 1, and D is the forward difference operator D f (x) = f (x + 1) − f (x).
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