Special classes of polynomials [lecture notes]

219 Pages·2014·0.835 MB·English

by  Gospava B. Djordjevic, Gradimir V. Milovanovic

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