Special classes of polynomials Gospava B. Djordjevi´c Gradimir V. Milovanovi´c University of Niˇs, Faculty of Technology Leskovac, 2014. ii Preface In this book we collect several recent results on special classes of polynomi- als. We mostlyfocustoclassesofpolynomialsrelated toclassicalorthogonal polynomials. These classes are named as polynomials of Legendre, Gegen- bauer, Chebyshev, Hermite, Laguerre, Jacobsthal, Jacobsthal – Lucas, Fi- bonacci, Pell, Pell – Lucas, Morgan – Voyce. Corresponding numbers are frequently investigated. We present new relations, explicit representations and generating functions. We are not able to collect all results in this topic, so we reduce material to subjects of our own interest. Authors Leskovac and Belgrade, 2014. iii iv Contents 1 Standard classes of polynomials 1 1.1 Bernoulli and Euler polynomials . . . . . . . . . . . . . . . . 1 1.1.1 Introductory remarks . . . . . . . . . . . . . . . . . . 1 1.1.2 Bernoulli polynomials . . . . . . . . . . . . . . . . . . 2 1.1.3 Euler polynomials . . . . . . . . . . . . . . . . . . . . 7 1.1.4 Zeros of Bernoulli and Euler polynomials . . . . . . . 9 1.1.5 Real zeros of Bernoulli polynomials . . . . . . . . . . . 12 1.2 Orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . 14 1.2.1 Moment–functional and orthogonality . . . . . . . . . 14 1.2.2 Properties of orthogonal polynomials . . . . . . . . . . 16 1.2.3 The Stieltjes procedure . . . . . . . . . . . . . . . . . 18 1.2.4 Classical orthogonal polynomials . . . . . . . . . . . . 19 1.2.5 Generating function . . . . . . . . . . . . . . . . . . . 23 1.3 Gegenbauer polynomials and generalizations . . . . . . . . . . 25 1.3.1 Properties of Gegenbauer polynomials . . . . . . . . . 25 1.3.2 Generalizations of Gegenbauer polynomials . . . . . . 27 1.3.3 The differential equation . . . . . . . . . . . . . . . . 34 1.3.4 Polynomials in a parameter λ . . . . . . . . . . . . . 37 1.3.5 Polynomials induced by polynomials pλ (x) . . . . . 38 n,m 1.3.6 Special cases . . . . . . . . . . . . . . . . . . . . . . . 41 1.3.7 Distribution of zeros . . . . . . . . . . . . . . . . . . . 42 1.3.8 Generalizations of Dilcher polynomials . . . . . . . . . 43 1.3.9 Special cases . . . . . . . . . . . . . . . . . . . . . . . 47 2 Horadam polynomials 49 2.1 Horadam polynomials . . . . . . . . . . . . . . . . . . . . . . 49 2.1.1 Introductory remarks . . . . . . . . . . . . . . . . . . 49 2.1.2 Pell and Pell–Lucas polynomials . . . . . . . . . . . . 51 2.1.3 Convolutions of Pell and Pell–Lucas polynomials . . . 53 v vi CONTENTS 2.1.4 Generalizations of the Fibonacci and Lucas polynomials . . . . . . . . . . . . . . . . . . . . . . . . 56 (k) 2.1.5 Polynomials U (x) . . . . . . . . . . . . . . . . . . . 57 n,4 (k) 2.1.6 Polynomials U (x) . . . . . . . . . . . . . . . . . . 61 n,m 2.1.7 Some interesting identities . . . . . . . . . . . . . . . 63 2.1.8 The sequence {C (r)} . . . . . . . . . . . . . . . . . 69 n,3 2.1.9 The sequence {C (r)} . . . . . . . . . . . . . . . . . 75 n,4 2.2 Pell, Pell–Lucas and Fermat polynomials . . . . . . . . . . . . 80 2.2.1 Polynomials Pk (x) and Qk (x) . . . . . . . . . . . 80 n,m n,m 2.2.2 Mixed convolutions . . . . . . . . . . . . . . . . . . . . 83 2.2.3 Generalizations of Fermat polynomials . . . . . . . . . 86 2.2.4 Numerical sequences . . . . . . . . . . . . . . . . . . . 91 3 Morgan–Voyce and Jacobsthal 95 3.1 Morgan–Voyce polynomials . . . . . . . . . . . . . . . . . . . 95 3.1.1 Introductory remarks . . . . . . . . . . . . . . . . . . 95 3.1.2 Polynomials U (p,q;x) . . . . . . . . . . . . . . . . . 96 n,2 3.1.3 Polynomials U (p,q;x) . . . . . . . . . . . . . . . . 99 n,m 3.1.4 Particular cases . . . . . . . . . . . . . . . . . . . . . . 102 3.1.5 Diagonal polynomials . . . . . . . . . . . . . . . . . . 104 3.1.6 Generalizations of Morgan–Voyce polynomials . . . . . 107 (r) (r) 3.1.7 Polynomials P (x) and Q (x) . . . . . . . . . . . 107 n,m n,m 3.2 Jacobsthal polynomials. . . . . . . . . . . . . . . . . . . . . . 113 3.2.1 Introductory remarks . . . . . . . . . . . . . . . . . . 113 3.2.2 Polynomials J (x) and j (x) . . . . . . . . . . . . 114 n,m n,m 3.2.3 Polynomials F (x) and f (x) . . . . . . . . . . . 116 n,m n,m 3.2.4 Polynomials related to generalized Chebyshev polynomials . . . . . . . . . . . . . . . . . 123 3.2.5 Polynomials P (x) and Chebyshev polynomials . . . 125 n,3 3.2.6 General polynomials . . . . . . . . . . . . . . . . . . . 127 3.2.7 Chebyshev and Jacobsthal polynomials . . . . . . . . 129 3.2.8 Mixed convolutions of the Chebyshev type . . . . . . 132 3.2.9 Incomplete generalized Jacobsthal and Jacobsthal–Lucas numbers . . . . . . . . . . . . . 134 4 Hermite and Laguerre polynomials 143 4.1 Generalized Hermite polynomials . . . . . . . . . . . . . . . . 143 4.1.1 Introductory remarks . . . . . . . . . . . . . . . . . . 143 4.1.2 Properties of polynomials hλ (x) . . . . . . . . . . . 144 n,m CONTENTS vii 4.1.3 Polynomials with two parameters . . . . . . . . . . . 146 4.1.4 Generalized polynomials with the parameter λ . . . . 148 4.1.5 Special cases and distribution of zeros . . . . . . . . . 151 4.1.6 The Rodrigues type formula . . . . . . . . . . . . . . . 153 4.1.7 Special cases . . . . . . . . . . . . . . . . . . . . . . . 154 4.1.8 The operator formula . . . . . . . . . . . . . . . . . . 156 4.1.9 Implications related to generalized polynomials. . . . 157 4.2 Polynomials induced by generalized Hermite . . . . . . . . . . 160 4.2.1 Polynomials with two parameters . . . . . . . . . . . . 160 4.2.2 Distribution of zeros . . . . . . . . . . . . . . . . . . . 163 4.2.3 Polynomials related to the generalized Hermite polynomials . . . . . . . . . . . . . . . . . . . 165 4.2.4 Explicit formulas for the polynomials heν (z,x;α,β) 169 n,m 4.2.5 Polynomials {Hm(λ)} . . . . . . . . . . . . . . . . . . 170 n 4.2.6 Connection of the polynomials {Hm(λ)} and n the hyperbolic functions . . . . . . . . . . . . . . . . . 173 4.2.7 Polynomials {Hm (λ)} . . . . . . . . . . . . . . . . . 175 r,n 4.2.8 A natural generating function . . . . . . . . . . . . . 177 4.2.9 A conditional generating function . . . . . . . . . . . 178 4.3 Generalizations of Laguerre polynomials . . . . . . . . . . . . 181 4.3.1 Introductory remarks . . . . . . . . . . . . . . . . . . 181 4.3.2 Polynomials (cid:96)s (x) . . . . . . . . . . . . . . . . . . . 183 n,m 4.3.3 Generalization of Panda polynomials . . . . . . . . . . 185 4.3.4 Polynomials ga (x) and ha (x) . . . . . . . . . . . . 188 n,m n,m 4.3.5 Convolution type equalities . . . . . . . . . . . . . . . 190 4.3.6 Polynomials of the Laguerre type . . . . . . . . . . . . 193 4.3.7 Polynomials fc,r (x) . . . . . . . . . . . . . . . . . . . 194 n,m 4.3.8 Some special cases of fc,r (x) . . . . . . . . . . . . . . 196 n,m 4.3.9 Some identities of the convolution type . . . . . . . . 200 viii CONTENTS Chapter 1 Standard classes of polynomials 1.1 Bernoulli and Euler polynomials 1.1.1 Introductory remarks In 1713 Jacob Bernoulli introduced one infinite sequence of numbers in an elementary way. Bernoulli’s results appeared in his work “Ars conjectandi” forthefirsttime. ThesenumbersareknownasBernoullinumbers. Bernoulli investigated sums of the form S (n) = 1p+2p+3p+···+np. p He obtained the result that these numbers can be written in the form of the following polynomials (cid:18) (cid:19) (cid:18) (cid:19) 1 1 1 p 1 p S (n) = np+1+ np+ Anp−1+ Bnp−3+..., p p+1 2 2 1 4 3 whose coefficients contain the sequence of rational numbers 1 1 1 1 A = , B = − , C = , D = − ,.... 6 30 42 30 Later, Euler [23] investigated the same problem independently of Ber- noulli. Euler also introduced the sequence of rational numbers A, B, C, D, ... . In the work “Introductio in analusin infinitorum”, 1748, Euler noticed the connection between infinite sums 1 1 1 s(2n) = + + +··· = α π2n 12n 22n 32n n 1 2 CHAPTER 1. STANDARD CLASSES OF POLYNOMIALS and rational coefficients α which contain the same sequence of numbers A, n B, C, D, ... . Euler [23] obtained interesting results, showing that these numbers are contained in coefficients of the series expansion of functions x (cid:55)→ cotx, x (cid:55)→ tanx, x (cid:55)→ 1/sinx. Euler admitted Bernoulli’s priority in this subject and named these rational numbers as Bernoulli numbers. It was shown that these numbers have lots of applications, so they became the subject of study of many mathematicians (Jacobi, Carlitz [9], Delange [13], Dilcher [17], [18] , [19], Rakoˇcevi´c [97], etc.). In the same time properties and applications of Bernoulli polynomials S (n)areinvestigated. WecansaythatBernoullipolynomialsformaspecial p classofpolynomialsbecauseoftheirgreatapplicability. Themostimportant applications of these polynomials are in theory of finite differences, analytic number theory and lots of applications in classical analysis. 1.1.2 Bernoulli polynomials The coefficient B of the Teylor expansion of the function t (cid:55)→ g(t) n = t/(et−1), i.e., ∞ g(t) = t = (cid:88) Bntn. (1.2.1) et−1 n! n=0 Numbers B are rational. Namely, we have n 1 1 1 1 B = 1, B = − , B = , B = − , B = , ..., 0 1 2 4 6 2 6 30 42 and B = 0, for all k ≥ 1. 2k+1 Remark 1.1.1. Bernoulli numbers B can be expressed by the following n Euler’s formula 2(2k)! B = (−1)k+1 ζ(2k) (1.2.2) 2k (2π)2k where ∞ (cid:88) 1 ζ(z) = ((cid:60)(z) > 1) nz n=1 is the Riemann zeta function. Thus, according to (1.2.2), we can conclude that the following holds: (−1)k+1B > 0 for all k ≥ 1. 2k