Table of Contents
ISRN Algebra
Volume 2011, Article ID 268096, 18 pages
http://dx.doi.org/10.5402/2011/268096
Research Article

Sequences of Non-Gegenbauer-Humbert Polynomials Meet the Generalized Gegenbauer-Humbert Polynomials

1Department of Mathematics and Computer Science, Illinois Wesleyan University, Bloomington, IL 61702, USA
2Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV 89154-4020, USA

Received 19 April 2011; Accepted 12 May 2011

Academic Editors: F. Kittaneh, L. Vinet, and A. Vourdas

Copyright © 2011 Tian-Xiao He and Peter J.-S. Shiue. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Here, we present a connection between a sequence of polynomials generated by a linear recurrence relation of order 2 and sequences of the generalized Gegenbauer-Humbert polynomials. Many new and known transfer formulas between non-Gegenbauer-Humbert polynomials and generalized Gegenbauer-Humbert polynomials are given. The applications of the relationship to the construction of identities of polynomial sequences defined by linear recurrence relations are also discussed.

1. Introduction

Many number and polynomial sequences can be defined, characterized, evaluated, and classified by linear recurrence relations with certain orders. A polynomial sequence {𝑎𝑛(𝑥)} is called a sequence of order 2 if it satisfies the linear recurrence relation of order 2𝑎𝑛(𝑥)=𝑝(𝑥)𝑎𝑛1+𝑞(𝑥)𝑎𝑛2(𝑥),𝑛2,(1.1) for some coefficient 𝑝(𝑥)0 and 𝑞(𝑥)0 and initial conditions 𝑎0(𝑥) and 𝑎1(𝑥). To construct an explicit formula of its general term, one may use a generating function, characteristic equation, or a matrix method (see Comtet [1], Hsu [2], Strang [3], Wilf [4], etc.). In [5], the authors presented a new method to construct an explicit formula of {𝑎𝑛(𝑥)} generated by (1.1). For the sake of the reader's convenience, we cite this result as follows.

Proposition 1.1. Let {𝑎𝑛(𝑥)} be a sequence of order 2 satisfying the linear recurrence relation (1.1), then 𝑎𝑛𝑎(𝑥)=1(𝑥)𝛽(𝑥)𝑎0(𝑥)𝛼𝛼(𝑥)𝛽(𝑥)𝑛𝑎(𝑥)1(𝑥)𝛼(𝑥)𝑎0(𝑥)𝛽𝛼(𝑥)𝛽(𝑥)𝑛(𝑥)if𝛼(𝑥)𝛽(𝑥),𝑛𝑎1(𝑥)𝛼𝑛1(𝑥)(𝑛1)𝑎0(𝑥)𝛼𝑛(𝑥)if𝛼(𝑥)=𝛽(𝑥),(1.2) where 𝛼(𝑥) and 𝛽(𝑥) are roots of 𝑡2𝑝(𝑥)𝑡𝑞(𝑥)=0, namely, 1𝛼(𝑥)=2𝑝(𝑥)+𝑝21(𝑥)+4𝑞(𝑥),𝛽(𝑥)=2𝑝(𝑥)𝑝2.(𝑥)+4𝑞(𝑥)(1.3)

In [6], Aharonov et al. have proved that the solution of any sequence of numbers that satisfies a recurrence relation of order 2 with constant coefficients and initial conditions 𝑎0=0 and 𝑎1=1, called the primary solution, can be expressed in terms of Chebyshev polynomial values. For instance, the authors show 𝐹𝑛=𝑖𝑛𝑈𝑛(𝑖/2)and 𝐿𝑛=2𝑖𝑛𝑇𝑛(𝑖/2), where 𝐹𝑛 and 𝐿𝑛 are, respectively, Fibonacci numbers and Lucas numbers, and 𝑇𝑛(𝑥) and 𝑈𝑛(𝑥) are the Chebyshev polynomials of the first kind and the second kind, respectively. Some identities drawn from those relations were given by Beardon in [7]. Marr and Vineyard in [8] use the relationship to establish explicit expression of five-diagonal Toeplitz determinants. In [5], the authors presented a new method to construct an explicit formula of {𝑎𝑛(𝑥)} generated by (1.1). Inspired with those results, in [9], The authors and Weng established a relationship between the number sequences defined by recurrence relation (1.1) and the generalized Gegenbauer-Humbert polynomial value sequences. The results are suitable for all such number sequences defined by (1.1) with arbitrary initial conditions 𝑎0 and 𝑎1, which includes the results in [6, 7] as the special cases. Many new and known formulas of Fibonacci, Lucas, Pell, and Jacobsthal numbers in terms of the generalized Gegenbauer-Humbert polynomial values were presented in [9]. In this paper, we will give an alternative form of (1.2) and find a relationship between all polynomial sequences defined by (1.1) and the generalized Gegenbauer-Humbert polynomial sequences.

A sequence of the generalized Gegenbauer-Humbert polynomials {𝑃𝑛𝜆,𝑦,𝐶(𝑥)}𝑛0 is defined by the expansion (see, e.g., [1], Gould [10], and the authors with Hsu [11]) Φ(𝑡)𝐶2𝑥𝑡+𝑦𝑡2𝜆=𝑛0𝑃𝑛𝜆,𝑦,𝐶(𝑥)𝑡𝑛,(1.4) where 𝜆>0, 𝑦 and 𝐶0 are real numbers. As special cases of (1.4), we consider 𝑃𝑛𝜆,𝑦,𝐶(𝑥) as follows (see [11]): 𝑃𝑛1,1,1(𝑥)=𝑈𝑛(𝑥),Chebyshev polynomial of the second kind, 𝑃𝑛1/2,1,1(𝑥)=𝜓𝑛(𝑥), Legendre polynomial, 𝑃𝑛1,1,1(𝑥)=𝑃𝑛+1(𝑥), Pell polynomial,𝑃𝑛1,1,1(𝑥/2)=𝐹𝑛+1(𝑥), Fibonacci polynomial,𝑃𝑛1,1,1((𝑥/2)+1)=𝐵𝑛(𝑥), Morgan-Voyc polynomial, [12] by Koshy, 𝑃𝑛1,2,1(𝑥/2)=Φ𝑛+1(𝑥), Fermat polynomial of the first kind,𝑃𝑛1,2𝑎,2(𝑥)=𝐷𝑛(𝑥,𝑎), Dickson polynomial of the second kind,𝑎0 (see, e.g., [13]) by Lidl et al.,

where 𝑎 is a real parameter, and 𝐹𝑛=𝐹𝑛(1) is the Fibonacci number. In particular, if 𝑦=𝐶=1, the corresponding polynomials are called Gegenbauer polynomials (see [1]). More results on the Gegenbauer-type polynomials can be found in Hsu [14] and Hsu and Shiue [15], and so forth, it is interesting that for each generalized Gegenbauer-Humbert polynomial sequence there exists a nongeneralized Gegenbauer-Humbert polynomial sequence, for instance, corresponding to the Chebyshev polynomials of the second kind, Pell polynomials, Fibonacci polynomials, Fermat polynomials of the first kind, and the Dickson polynomials of the second kind, we have the Chebyshev polynomials of the first kind, Pell-Lucas polynomials (see [16] by Horadam and Mahon), Lucas polynomials, the Fermat polynomials of the second kind (see [17] by Horadam), and the Dickson polynomials of the first kind, respectively.

Similarly, for a class of the generalized Gegenbauer-Humbert polynomial sequences defined by 𝑃𝑛𝜆,𝑦,𝐶(𝑥)=2𝑥𝜆+𝑛1𝑃𝐶𝑛𝜆,𝑦,𝐶𝑛1(𝑥)𝑦2𝜆+𝑛2𝑃𝐶𝑛𝜆,𝑦,𝐶𝑛2(𝑥),(1.5) for all 𝑛2 with initial conditions 𝑃0𝜆,𝑦,𝐶(𝑥)=Φ(0)=𝐶𝜆,𝑃1𝜆,𝑦,𝐶(𝑥)=Φ(0)=2𝜆𝑥𝐶𝜆1,(1.6) the following theorem is obtained.

Theorem 1.2 (see [5]). Let 𝑥±𝐶𝑦. The generalized Gegenbauer-Humbert polynomials {𝑃𝑛1,𝑦,𝐶(𝑥)}𝑛0 defined by expansion (1.4) can be expressed as 𝑃𝑛1,𝑦,𝐶(𝑥)=𝐶𝑛2𝑥+𝑥2𝐶𝑦𝑛+1𝑥𝑥2𝐶𝑦𝑛+12𝑥2.𝐶𝑦(1.7)

In next section, we will use an alternative form of (1.2) to establish a relationship between the polynomial sequences defined by recurrence relation (1.1) and the generalized Gegenbauer-Humbert polynomial sequences defined by (1.5). Many new and known formulas of polynomials in terms of the generalized Gegenbauer-Humbert polynomials and applications of the established relationship to the construction of identities of polynomial sequences will be presented in Section 3.

2. Main Results

We now modify the explicit formula of the polynomial sequences defined by linear recurrence relation (1.2) of order 2. If 𝛼(𝑥)𝛽(𝑥), the first formula in (1.2) can be written as 𝑎𝑛(𝑎𝑥)=1(𝑥)((𝛼(𝑥))𝑛(𝛽(𝑥))𝑛)𝑎0(𝑥)𝛼(𝑥)𝛽(𝑥)(𝛼(𝑥))𝑛1(𝛽(𝑥))𝑛1.𝛼(𝑥)𝛽(𝑥)(2.1) Noting that 𝛼(𝑥)𝛽(𝑥)=𝛼(𝑥)(𝛼(𝑥)𝑝(𝑥))=𝛽(𝑥)(𝛽(𝑥)𝑝(𝑥)), we may further write the above expression of 𝑎𝑛(𝑥) as 𝑎𝑛1(𝑥)=𝛼𝑎(𝑥)𝛽(𝑥)1(𝑥)((𝛼(𝑥))𝑛(𝛽(𝑥))𝑛)+𝑎0(𝑥)𝛼(𝑥)(𝛼(𝑥)𝑝(𝑥))×(𝛼(𝑥))𝑛1𝑎0(𝑥)𝛽(𝑥)(𝛽(𝑥)𝑝(𝑥))(𝛽(𝑥))𝑛1=𝑎0(𝑥)(𝛼(𝑥))𝑛+1(𝛽(𝑥))𝑛+1+𝑎1(𝑥)𝑎0(𝑥)𝑝(𝑥)((𝛼(𝑥))𝑛(𝛽(𝑥))𝑛)𝛼.(𝑥)𝛽(𝑥)(2.2)

Denote 𝑟(𝑥)=𝑥+𝑥2𝐶𝑦 and 𝑠(𝑥)=𝑥𝑥2𝐶𝑦. To find a transfer formula between expressions (1.7) and (2.2), we set 𝛼(𝑥)=𝑟(𝑥)𝑘(𝑥),𝛽(𝑥)=𝑠(𝑥)𝑘(𝑥),(2.3) for a nonzero real or complex-valued function 𝑘(𝑥), which are two roots of 𝑡2𝑝(𝑥)𝑡𝑞(𝑥)=0. Thus, adding and multiplying two equations of (2.3) side by side, we obtain 𝛼(𝑥)+𝛽(𝑥)=𝑝(𝑥)=2𝑥𝑘,(𝑥)𝛼(𝑥)𝛽(𝑥)=𝑞(𝑥)=𝐶𝑦(𝑘(𝑥))2.(2.4) The above system implies 𝑘(𝑥)=±𝐶𝑦,𝑞(𝑥)(2.5) and at 𝑥=𝑝(𝑥)𝑘(𝑥)2=±𝑝(𝑥)2𝐶𝑦,𝑞(𝑥)(2.6)𝑟(𝑥) and 𝑠(𝑥) give expressions of 𝛼(𝑥) and 𝛽(𝑥) as 𝑟𝛼(𝑥)=±(𝑝(𝑥)/2)𝐶𝑦/𝑞(𝑥)±𝑠𝐶𝑦/𝑞(𝑥),𝛽(𝑥)=±(𝑝(𝑥)/2)𝐶𝑦/𝑞(𝑥)±𝐶𝑦/𝑞(𝑥).(2.7) It is clear that 𝛼(𝑥) and 𝛽(𝑥) satisfy 𝛼(𝑥)+𝛽(𝑥)=𝑝(𝑥) and 𝛼(𝑥)𝛽(𝑥)=𝑞(𝑥).

We first consider the case of 𝑘(𝑥)=𝐶𝑦/𝑞(𝑥). Substituting the corresponding (2.7) with positive sign into (2.2), we have 𝑎𝑛𝑎(𝑥)=0𝑟(𝑥)𝑛+1(𝑥)𝑠𝑛+1𝑎(𝑥)+𝑘(𝑥)1(𝑥)𝑎0(𝑥)𝑝(𝑥)(𝑟𝑛(𝑥)𝑠𝑛(𝑥))𝑘𝑛(𝑥)(𝑟(𝑥)𝑠(𝑥))=𝑎0(𝑥)𝐶𝑛+2𝑞(𝑥)𝐶𝑦𝑛𝑃𝑛1,𝑦,𝐶𝑘(𝑥)𝑝(𝑥)2+𝑎1(𝑥)𝑎0𝐶(𝑥)𝑝(𝑥)𝑛+1𝑞(𝑥)𝐶𝑦𝑛1𝑃1,𝑦,𝐶𝑛1𝑘(𝑥)𝑝(𝑥)2=𝑎0(𝑥)𝐶𝑛+2𝑞(𝑥)𝐶𝑦𝑛𝑃𝑛1,𝑦,𝐶𝑝(𝑥)2𝐶𝑦+𝑎𝑞(𝑥)1(𝑥)𝑎0𝐶(𝑥)𝑝(𝑥)𝑛+1𝑞(𝑋)𝐶𝑦𝑛1𝑃1,𝑦,𝐶𝑛1𝑝(𝑥)2𝐶𝑦.𝑞(𝑥)(2.8) Similarly, for 𝑘(𝑥)=𝐶𝑦/𝑞(𝑥), we have 𝑎𝑛(𝑥)=𝑎0(𝑥)𝐶𝑛+2𝑞(𝑥)𝐶𝑦𝑛𝑃𝑛1,𝑦,𝐶𝑝(𝑥)2𝐶𝑦+𝑎𝑞(𝑥)1(𝑥)𝑎0𝐶(𝑥)𝑝(𝑥)𝑛+1𝑞(𝑥)𝐶𝑦𝑛1𝑃1,𝑦,𝐶𝑛1𝑝(𝑥)2𝐶𝑦.𝑞(𝑥)(2.9)

Therefore, we obtain our main result.

Theorem 2.1. Let sequence {𝑎𝑛(𝑥)}𝑛0 be defined by 𝑎𝑛(𝑥)=𝑝(𝑥)𝑎𝑛1(𝑥)+𝑞(𝑥)𝑎𝑛2(𝑥)(𝑛2) with initial conditions 𝑎0(𝑥) and 𝑎1(𝑥), then 𝑎𝑛(𝑥) can be presented as 𝑎𝑛(𝑥)=𝑎0(𝑥)𝐶𝑛+2±𝑞(𝑥)𝐶𝑦𝑛𝑃𝑛1,𝑦,𝐶±𝑝(𝑥)2𝐶𝑦+𝑎𝑞(𝑥)1(𝑥)𝑎0𝐶(𝑥)𝑝(𝑥)𝑛+1±𝑞(𝑋)𝐶𝑦𝑛1𝑃1,𝑦,𝐶𝑛1±𝑝(𝑥)2𝐶𝑦,𝑞(𝑥)(2.10) where {𝑃𝑛1,𝑦,𝑐} is the sequence of any generalized Gegenbauer-Humbert polynomials with 𝜆=1. In particular, 𝑎𝑛(𝑥) can be expressed in terms of {𝑃𝑛1,1,1=𝑈𝑛}, the sequence of the Chebyshev polynomials of the second kind, 𝑎𝑛(𝑥)=𝑎0±(𝑥)𝑞(𝑥)𝑛𝑈𝑛±𝑝(𝑥)2+𝑎𝑞(𝑥)1(𝑥)𝑎0±(𝑥)𝑝(𝑥)𝑞(𝑥)𝑛1𝑈𝑛1±𝑝(𝑥)2,𝑞(𝑥)(2.11) which is a special case of (2.10) for (𝑦,𝐶)=(1,1).

Corollary 2.2. For (𝑦,𝐶)=(1,1),(1,1),(2,1),𝑎𝑛𝑑(2𝑎,2)(𝑎0), respectively, from (2.10), one has transfer formulas 𝑎𝑛(𝑥)=𝑎0±(𝑥)𝑞(𝑥)𝑛𝑃𝑛+1±𝑝(𝑥)2+𝑎𝑞(𝑥)1(𝑥)𝑎0±(𝑥)𝑝(𝑥)𝑞(𝑥)𝑛1𝑃𝑛±𝑝(𝑥)2,𝑎𝑞(𝑥)𝑛(𝑥)=𝑎0±(𝑥)𝑞(𝑥)𝑛𝐹𝑛+1±𝑝(𝑥)𝑞+𝑎(𝑥)1(𝑥)𝑎0±(𝑥)𝑝(𝑥)𝑞(𝑥)𝑛1𝐹𝑛±𝑝(𝑥),𝑎𝑞(𝑥)𝑛(𝑥)=𝑎0±(𝑥)𝑞(𝑥)𝑛𝐵𝑛±𝑝(𝑥)+𝑎𝑞(𝑥)21(𝑥)𝑎0±(𝑥)𝑝(𝑥)𝑞(𝑥)𝑛1𝐵𝑛1±𝑝(𝑥),𝑎𝑞(𝑥)2𝑛(𝑥)=𝑎0±(𝑥)𝑞(𝑥)2𝑛Φ𝑛+1±𝑝(𝑥)2+𝑎𝑞(𝑥)1(𝑥)𝑎0±(𝑥)𝑝(𝑥)𝑞(𝑥)2𝑛1Φ𝑛±𝑝(𝑥)2,𝑎𝑞(𝑥)𝑛(𝑥)=4𝑎0±(𝑥)𝑞(𝑥)𝑎𝑛𝐷𝑛±𝑝(𝑥)𝑎𝑎𝑞(𝑥),𝑎+41(𝑥)𝑎0±(𝑥)𝑝(𝑥)𝑞(𝑥)𝑎𝑛1𝐷𝑛1±𝑝(𝑥)𝑎,𝑞(𝑥),𝑎(2.12) where 𝑈𝑛(𝑥), 𝑃𝑛(𝑥), 𝐹𝑛(𝑥), Φ𝑛(𝑥), and 𝐷𝑛(𝑥,𝑎) are the Chebyshev polynomials of the second order, Pell polynomials, Fibonacci polynomials, Fermat polynomials, and the Dickson polynomials of the second kind, respectively.

Example 2.3. As the first example, we consider the Chebyshev polynomials of the first kind 𝑇𝑛(𝑥)=cos(𝑛arccos𝑥) satisfying recurrence relation (1.1) with 𝑝(𝑥)=2𝑥 and 𝑞=1 and initial conditions 𝑇0(𝑥)=1 and 𝑇1(𝑥)=𝑥. From Corollary 2.2, we have 𝑇𝑛(𝑥)=𝑈𝑛(𝑥)𝑥𝑈𝑛1𝑇(𝑥),𝑛(𝑥)=(1)𝑛𝑈𝑛(𝑥)+𝑥𝑈𝑛1,𝑇(𝑥)𝑛(𝑥)=(±𝑖)𝑛𝑃𝑛+1(𝑥𝑖)𝑥(±𝑖)𝑛1𝑃𝑛𝑇(𝑥𝑖),𝑛(𝑥)=(±𝑖)𝑛𝐹𝑛+1(2𝑥𝑖)𝑥(±𝑖)𝑛1𝐹𝑛𝑇(2𝑥𝑖),𝑛(𝑥)=(±1)𝑛𝐵𝑛(±2𝑥2)(±1)𝑛1𝑥𝐵𝑛1𝑇(±2𝑥2),𝑛(𝑥)=𝐵𝑛(±2𝑥2)𝑥𝐵𝑛1(𝑇±2𝑥2),𝑛±1(𝑥)=2𝑛Φ𝑛+12±12𝑥𝑥2𝑛1Φ𝑛2,𝑇2𝑥𝑛±1(𝑥)=4𝑎𝑛𝐷𝑛2±1𝑎𝑥,𝑎𝑥4𝑎𝑛1𝐷𝑛12,𝑎𝑥,𝑎(2.13) in which the first relation is equivalent to the well-known result 2𝑇𝑛(𝑥)=𝑈𝑛(𝑥)𝑈𝑛2(𝑥) due to 2𝑇𝑛(𝑥)=2𝑈𝑛(𝑥)2𝑥𝑈𝑛1(𝑥)=𝑈𝑛(𝑥)+2𝑥𝑈𝑛1(𝑥)𝑈𝑛2(𝑥)2𝑥𝑈𝑛1(𝑥).(2.14)
For the special cases of 𝑎0(𝑥) and 𝑎1(𝑥), we have the following corollaries.

Corollary 2.4. Let sequence {𝑎𝑛(𝑥)}𝑛0 be defined by 𝑎𝑛(𝑥)=𝑝(𝑥)𝑎𝑛1(𝑥)+𝑞(𝑥)𝑎𝑛2(𝑥) (𝑛2) with initial conditions 𝑎0(𝑥)=0 and 𝑎1(𝑥)=𝑑. Then 𝑎𝑛±(𝑥)=𝑑𝑞(𝑥)𝑛1𝑈𝑛1±𝑝(𝑥)2,𝑎𝑞(𝑥)𝑛±(𝑥)=𝑑𝑞(𝑥)𝑛1𝑃𝑛±𝑝(𝑥)2,𝑎𝑞(𝑥)𝑛±(𝑥)=𝑑𝑞(𝑥)𝑛1𝐹𝑛±𝑝(𝑥)𝑞,𝑎(𝑥)𝑛±(𝑥)=𝑑𝑞(𝑥)𝑛1𝐵𝑛1±𝑝(𝑥),𝑎𝑞(𝑥)2𝑛±(𝑥)=𝑑𝑞(𝑥)2𝑛1Φ𝑛±𝑝(𝑥)2,𝑎𝑞(𝑥)𝑛±(𝑥)=4𝑑𝑞(𝑥)𝑎𝑛1𝐷𝑛1±𝑝(𝑥)𝑎.𝑞(𝑥),𝑎(2.15)

Corollary 2.5. Let sequence {𝑎𝑛(𝑥)}𝑛0 be defined by 𝑎𝑛(𝑥)=𝑝(𝑥)𝑎𝑛1(𝑥)+𝑞(𝑥)𝑎𝑛2(𝑥)(𝑛2) with initial conditions 𝑎0(𝑥)=𝑐 and 𝑎1(𝑥)=𝑐𝑝(𝑥), then 𝑎𝑛±(𝑥)=𝑐𝑞(𝑥)𝑛𝑈𝑛±𝑝(𝑥)2,𝑎𝑞(𝑥)𝑛±(𝑥)=𝑐𝑞(𝑥)𝑛𝑃𝑛+1±𝑝(𝑥)2,𝑎𝑞(𝑥)𝑛±(𝑥)=𝑐𝑞(𝑥)𝑛𝐹𝑛+1±𝑝(𝑥)𝑞,𝑎(𝑥)𝑛±(𝑥)=𝑐𝑞(𝑥)𝑛𝐵𝑛±𝑝(𝑥),𝑎𝑞(𝑥)2𝑛(±𝑥)=𝑐𝑞(𝑥)2𝑛Φ𝑛+1±𝑝(𝑥)2,𝑎𝑞(𝑥)𝑛±(𝑥)=4𝑐𝑞(𝑥)𝑎𝑛𝐷𝑛±𝑝(𝑥)𝑎.𝑞(𝑥),𝑎(2.16)

We now give another special case of Theorem 2.1 for the sequence defined by (1.1) with initial cases 𝑎0(𝑥)=2 and 𝑎1(𝑥)=𝑝(𝑥).

Corollary 2.6. Let sequence {𝑎𝑛(𝑥)}𝑛0 be defined by 𝑎𝑛(𝑥)=𝑝(𝑥)𝑎𝑛1(𝑥)+𝑞(𝑥)𝑎𝑛2(𝑥)(𝑛2) with initial conditions 𝑎0(𝑥)=2 and 𝑎1(𝑥)=𝑝(𝑥).
Then 𝑎𝑛±(𝑥)=2𝑞(𝑥)𝑛𝑈𝑛±𝑝(𝑥)2±𝑞(𝑥)𝑝(𝑥)𝑞(𝑥)𝑛1𝑈𝑛1±𝑝(𝑥)2,𝑎𝑞(𝑥)𝑛±(𝑥)=2𝑞(𝑥)𝑛𝑃𝑛+1±𝑝(𝑥)2𝑞±(𝑥)𝑝(𝑥)𝑞(𝑥)𝑛1𝑃𝑛±𝑝(𝑥)2,𝑎𝑞(𝑥)𝑛±(𝑥)=2𝑞(𝑥)𝑛𝐹𝑛+1±𝑝(𝑥)±𝑞(𝑥)𝑝(𝑥)𝑞(𝑥)𝑛1𝐹𝑛±𝑝(𝑥),𝑎𝑞(𝑥)𝑛±(𝑥)=2𝑞(𝑥)𝑛𝐵𝑛±𝑝(𝑥)±𝑞(𝑥)2𝑝(𝑥)𝑞(𝑥)𝑛1𝐵𝑛1±𝑝(𝑥),𝑎𝑞(𝑥)2𝑛±(𝑥)=2𝑞(𝑥)2𝑛Φ𝑛+1±𝑝(𝑥)2±𝑞(𝑥)𝑝(𝑥)𝑞(𝑥)2𝑛1Φ𝑛±𝑝(𝑥)2,𝑎𝑞(𝑥)𝑛(𝑥)=23±𝑞(𝑥)𝑎𝑛𝐷𝑛±𝑝(𝑥)𝑎𝑞(𝑥),𝑎𝑝(𝑥)22±𝑞(𝑥)𝑎𝑛1𝐷𝑛1±𝑝(𝑥)𝑎.𝑞(𝑥),𝑎(2.17) In addition, one has 𝑎𝑛±(𝑥)=2𝑞(𝑥)𝑛𝑇𝑛±𝑝(𝑥)2𝑞(𝑥),(2.18) where 𝑇𝑛(𝑥) are the Chebyshev polynomials of the first kind.

Proof. It is sufficient to prove the positive case of (2.18). From the first formula shown in Corollary 2.6 and the recurrence relation 𝑈𝑛(𝑥)=2𝑥𝑈𝑛1(𝑥)𝑈𝑛2(𝑥), one easily sees 𝑎𝑛(𝑥)=𝑞(𝑥)𝑛2𝑈𝑛𝑝(𝑥)2𝑞(𝑥)𝑝(𝑥)𝑈𝑞(𝑥)𝑛1𝑝(𝑥)2=𝑞(𝑥)𝑞(𝑥)𝑛2𝑈𝑛𝑝(𝑥)2𝑈𝑞(𝑥)𝑛𝑝(𝑥)2𝑞(𝑥)+𝑈𝑛2𝑝(𝑥)2=𝑞(𝑥)𝑞(𝑥)𝑛𝑈𝑛𝑝(𝑥)2𝑞(𝑥)𝑈𝑛2𝑝(𝑥)2.𝑞(𝑥)(2.19) From the first formula of Example 2.3, the above last expression of 𝑎𝑛(𝑥) implies the positive case of (2.18). The negative case can be proved similarly.

Example 2.7. As an example, the Lucas polynomial sequence {𝐿𝑛(𝑥)} defined by (1.1) with 𝑝(𝑥)=𝑥 and 𝑞(𝑥)=1 and initial conditions 𝐿0(𝑥)=2 and 𝐿1(𝑥)=𝑥 has an explicit formula for its general term 𝐿𝑛(𝑥)=2(±𝑖)𝑛𝑇𝑛𝑥𝑖2.(2.20) Using Corollary 2.6, we also have 𝐿𝑛(𝑥)=2(±𝑖)𝑛𝑈𝑛𝑥𝑖2𝑥(±𝑖)𝑛1𝑈𝑛1𝑥𝑖2,𝐿𝑛(𝑥)=2𝑃𝑛+1±𝑥2𝑥𝑃𝑛±𝑥2,𝐿𝑛(𝑥)=2𝐹𝑛+1(±𝑥)𝑥𝐹𝑛𝐿(±𝑥),𝑛(𝑥)=2(±𝑖)𝑛𝐵𝑛(𝑥𝑖2)𝑥(±𝑖)𝑛1𝐵𝑛1𝐿(𝑥𝑖2),𝑛±𝑖(𝑥)=22𝑛Φ𝑛+1±𝑖2𝑥𝑖𝑥2𝑛1Φ𝑛,𝐿2𝑥𝑖𝑛(𝑥)=23±𝑖𝑎𝑛𝐷𝑛𝑎𝑥𝑖,𝑎𝑥22±𝑖𝑎𝑛1𝐷𝑛1.𝑎𝑥𝑖,𝑎(2.21)
From Theorem 2.1, one may obtain transfer formulas between generalized Gegenbauer-Humbert polynomials.

3. Examples and Applications

We first give some examples of Theorem 2.1 for sequences {𝑎𝑛(𝑥)} defined by (1.1).

Example 3.1. The Chebyshev polynomials of the third kind and fourth kind satisfy the same recurrence relationship as the Chebyshev polynomials of the first kind with the same constant initial term 1 and different linear initial terms, 2𝑥1 and 2𝑥+1, respectively (see, e.g., Mason and Handscomb [18] and Rivlin [19]). Hence, the Chebyshev polynomials of the third kind, 𝑇𝑛(3)(𝑥), and the Chebyshev polynomials of the fourth kind, 𝑇𝑛(4)(𝑥), when 𝑥21, have the following expressions using the argument shown in [5]: 𝑇𝑛(3)(𝑥)=𝑥21+𝑥12𝑥21𝑥+𝑥21𝑛+𝑥21𝑥+12𝑥21𝑥𝑥21𝑛,𝑇𝑛(4)(𝑥)=𝑥21+𝑥+12𝑥21𝑥+𝑥21𝑛+𝑥21𝑥12𝑥21𝑥𝑥21𝑛.(3.1)

Similarly to the Chebyshev polynomials of the first kind (see Example 2.3), we can transfer 𝑇𝑛(3)(𝑥) and 𝑇𝑛(4)(𝑥) to the generalized Gegenbauer-Humbert polynomials with 𝜆=1, 𝑇𝑛(3)(𝑥)=𝑈𝑛(𝑥)𝑈𝑛1𝑇(𝑥),𝑛(3)(𝑥)=(1)𝑛𝑈𝑛(𝑥)+𝑈𝑛1(,𝑇𝑥)𝑛(3)(𝑥)=(±𝑖)𝑛𝑃𝑛+1(𝑥𝑖)(±𝑖)𝑛1𝑃𝑛𝑇(𝑥𝑖),𝑛(3)(𝑥)=(±𝑖)𝑛𝐹𝑛+1(2𝑥𝑖)(±𝑖)𝑛1𝐹𝑛𝑇(2𝑥𝑖),𝑛(3)(𝑥)=(±1)𝑛𝐵𝑛(±2𝑥2)(±1)𝑛1𝐵𝑛1𝑇(±2𝑥2),𝑛(3)±1(𝑥)=2𝑛Φ𝑛+12±12𝑥2𝑛1Φ𝑛2,𝑇2𝑥𝑛(3)±1(𝑥)=4𝑎𝑛𝐷𝑛2±1𝑎𝑥,𝑎4𝑎𝑛1𝐷𝑛12,𝑇𝑎𝑥,𝑎𝑛(4)(𝑥)=𝑈𝑛(𝑥)+𝑈𝑛1𝑇(𝑥),𝑛(4)(𝑥)=(1)𝑛𝑈𝑛(𝑥)𝑈𝑛1,𝑇(𝑥)𝑛(4)(𝑥)=(±𝑖)𝑛𝑃𝑛+1(𝑥𝑖)+(±𝑖)𝑛1𝑃𝑛𝑇(𝑥𝑖),𝑛(4)(𝑥)=(±𝑖)𝑛𝐹𝑛+1(2𝑥𝑖)+(±𝑖)𝑛1𝐹𝑛(𝑇2𝑥𝑖),𝑛(4)(𝑥)=(±1)𝑛𝐵𝑛(±2𝑥2)+(±1)𝑛1𝐵𝑛1𝑇(±2𝑥2),𝑛(4)±1(𝑥)=2𝑛Φ𝑛+12+±12𝑥2𝑛1Φ𝑛2,𝑇2𝑥𝑛(3)±1(𝑥)=4𝑎𝑛𝐷𝑛2+±1𝑎𝑥,𝑎4𝑎𝑛1𝐷𝑛12.𝑎𝑥,𝑎(3.2) From the above formulas, one may obtain some identities between the Chebyshev polynomials of different kinds. For instance, 𝑇𝑛(3)(𝑥)+𝑇𝑛(4)(𝑥)=2𝑈𝑛𝑇(𝑥),𝑛(𝑥)+𝑥𝑇𝑛(4)(𝑥)=(1+𝑥)𝑈𝑛(𝑇𝑥),𝑛(𝑥)𝑥𝑇𝑛(3)(𝑥)=(1𝑥)𝑈𝑛(𝑥).(3.3)

Since 𝑇𝑛(𝑥)=cos𝑛𝜃, 𝑈𝑛(𝑥)=sin(𝑛+1)𝜃/sin𝜃, 𝑇𝑛(3)(𝑥)=cos(𝑛+1/2)𝜃/cos(1/2)𝜃, and 𝑇𝑛(4)(𝑥)=sin(𝑛+1/2)𝜃/sin(1/2)𝜃, where 𝑥=cos𝜃, the above identities of Chebyshev polynomials also present the following identities of trigonometric functions, respectively, cos(𝑛+1/2)𝜃+cos(1/2)𝜃sin(𝑛+1/2)𝜃sin(1/2)𝜃=2sin(𝑛+1)𝜃,sin𝜃cos𝑛𝜃+cos𝜃sin(𝑛+1/2)𝜃=sin(1/2)𝜃(1+cos𝜃)sin(𝑛+1)𝜃,sin𝜃cos𝑛𝜃cos𝜃sin(𝑛+1/2)𝜃sin(1/2)𝜃=(1cos𝜃)sin(𝑛+1)𝜃.sin𝜃(3.4)

Example 3.2. Consider the Jacobsthal polynomials {𝐽𝑛(𝑥)} defined by (1.1) with coefficients 𝑝(𝑥)=1 and 𝑞(𝑥)=2𝑥 and initial conditions 𝐽0(𝑥)=𝐽1(𝑥)=1. One may use Corollary 2.5 to obtain transfer formulas 𝐽𝑛±(𝑥)=2𝑥𝑛𝑈𝑛±12,𝐽2𝑥𝑛±(𝑥)=2𝑥𝑛𝑃𝑛+1±12,𝐽2𝑥𝑛±(𝑥)=2𝑥𝑛𝐹𝑛+1±1,𝐽2𝑥𝑛±(𝑥)=2𝑥𝑛𝐵𝑛±1,𝐽2𝑥2𝑛±(𝑥)=𝑥𝑛Φ𝑛+1±1,𝐽𝑥𝑛(𝑥)=22±2𝑥𝑎𝑛𝐷𝑛±𝑎.2𝑥,𝑎(3.5) The first formula and its inverse (see the first formula below) were given on [20, page 76] by Riordan using a different method. The positive case of the third formula is easily to be transferred to the formula of Theorem  1 in [21], where they used a different recurrence relation with 𝑝(𝑥)=1 and 𝑞(𝑥)=𝑥 for constructing the Jacobsthal polynomials. Reference [20] also gave the inverse formula to present 𝑈𝑛(𝑥) in terms of 𝐽𝑛(𝑥). Actually, we can easily have the inverse formulas of 𝑈𝑛(𝑥), 𝑃𝑛+1(𝑥), 𝐹𝑛+1(𝑥), Φ𝑛+1(𝑥), and 𝐷𝑛(𝑥,𝑎) in terms of 𝐽𝑛(𝑥) as follows: 𝑈𝑛(𝑥)=(2𝑥)𝑛𝐽𝑛18𝑥2,𝑃𝑛+1(𝑥)=(2𝑥)𝑛𝐽𝑛18𝑥2,𝐹𝑛+1(𝑥)=𝑥𝑛𝐽𝑛12𝑥2,𝐵𝑛(𝑥)=(𝑥+2)𝑛𝐽𝑛12(𝑥+2)2,Φ𝑛+1(𝑥)=𝑥𝑛𝐽𝑛1𝑥2,𝐷𝑛1(𝑥,𝑎)=4𝑥𝑛𝐽𝑛𝑎2𝑥2.(3.6)

Example 3.3. In Eu [22], the polynomial sequence {𝐻𝑛(𝑥)} is defined by 𝑆𝑛(𝑥)=𝑥𝑆𝑛1(𝑥)𝑆𝑛2(𝑥) with initial conditions 𝑆0(𝑥)=1 and 𝑆1(𝑥)=𝑥. Using Corollary 2.5, we obtain 𝑆𝑛(𝑥)=𝑈𝑛±𝑥2,𝑆𝑛(𝑥)=(±𝑖)𝑛𝑃𝑛+1𝑥2𝑖,𝑆𝑛(𝑥)=(±𝑖)𝑛𝐹𝑛+1𝑆(𝑥𝑖),𝑛(𝑥)=(±1)𝑛𝐵𝑛𝑆(±𝑥2),𝑛(±1𝑥)=2𝑛Φ𝑛+1±,𝑆2𝑥𝑛±1(𝑥)=4𝑎𝑛𝐷𝑛±,𝑎𝑥,𝑎(3.7) in which the first formula was given in [22] using a different approach. Similar to the case of the Jacobsthal polynomial sequence shown in Example 3.2, we have the inverse formulas 𝑈𝑛(𝑥)=𝑆𝑛𝑃(±2𝑥),𝑛+1(𝑥)=(𝑖)𝑛𝑆𝑛𝐹(±2𝑥𝑖),𝑛+1(𝑥)=(𝑖)𝑛𝑆𝑛𝐵(±𝑥𝑖),𝑛(𝑥)=(±1)𝑛𝑆𝑛Φ(±(𝑥+2)),𝑛+1±(𝑥)=2𝑛𝑆𝑛±𝑥2,𝐷𝑛1(𝑥,𝑎)=4±𝑎𝑛𝑆𝑛±𝑥𝑎.(3.8)

Another polynomial sequence {𝐻𝑛(𝑥)} is defined by 𝐻𝑛(𝑥)=(1𝑥)𝐻𝑛1(𝑥)𝑥2𝐻𝑛2(𝑥) with initial conditions 𝐻0(𝑋)=1 and 𝐻1(𝑥)=1𝑥 [22]. Using Corollary 2.5, we obtain 𝐻𝑛(𝑥)=(±𝑥)𝑛𝑈𝑛±1𝑥,𝐻2𝑥𝑛(𝑥)=(±𝑖𝑥)𝑛𝑃𝑛+11𝑥𝑖,𝐻2𝑥𝑛(𝑥)=(±𝑖𝑥)𝑛𝐹𝑛+11𝑥𝑥𝑖,𝐻𝑛(𝑥)=(±𝑥)𝑛𝐵𝑛±1𝑥𝑥,𝐻2𝑛±𝑥(𝑥)=2𝑛Φ𝑛+1±21𝑥𝑥,𝐻𝑛±𝑥(𝑥)=4𝑎𝑛𝑈𝑛±𝑎1𝑥𝑥,,𝑎(3.9) where the first formula has been established in [22] by using a different method. The inverse of the above formulas can be found similarly. For instance, 𝑈𝑛(𝑥)=(2𝑥±1)𝑛𝐻𝑛11±2𝑥.(3.10)

Example 3.4. In Riordan [23], the associate Legendre polynomial sequence {𝜌𝑛(𝑥)} is defined by 𝜌𝑛(𝑥)=(2+𝑥)𝜌𝑛1(𝑥)𝜌𝑛2(𝑥) with initial conditions 𝜌0(𝑥)=1 and 𝜌1(𝑥)=1+𝑥, then we use Theorem 2.1 and Corollary 2.2 to generate the following transfer formulas: 𝜌𝑛(𝑥)=𝑈𝑛±𝑥1+2𝑈𝑛1±𝑥1+2,𝜌𝑛(𝑥)=(±𝑖)𝑛𝑃𝑛+1𝑥𝑖1+2(±𝑖)𝑛1𝑃𝑛𝑥𝑖1+2,𝜌𝑛(𝑥)=(±𝑖)𝑛𝐹𝑛+1(𝑖(𝑥+2))(±𝑖)𝑛1𝐹𝑛𝜌(𝑖(𝑥+2)),𝑛(𝑥)=(±1)𝑛𝐵𝑛(±(𝑥+2)2)(±1)𝑛1𝐵𝑛1𝜌(±(𝑥+2)2),𝑛(±1𝑥)=2𝑛Φ𝑛+1±±12(𝑥+2)2𝑛1Φ𝑛±,𝜌2(𝑥+2)𝑛±1(𝑥)=4𝑎𝑛𝐷𝑛±±1𝑎(𝑥+2),𝑎4𝑎𝑛1𝐷𝑛1±,𝑎(𝑥+2),𝑎(3.11) where the first formula was given on [20, page 85] using a different method.

Example 3.5. In Chow and West [24], the polynomial sequence {𝑝𝑛(𝑥)} is defined by 𝑝𝑛(𝑥)=𝑥𝑝𝑛1(𝑥)𝑥𝑝𝑛2(𝑥) with initial conditions 𝑝0(𝑥)=1𝑥1 and 𝑝1(𝑥)=2𝑥 (𝑥0). From Theorem 2.1 and Corollary 2.2, we obtain 𝑝𝑛(𝑥)=1𝑥1±𝑥𝑛𝑈𝑛𝑥2+±𝑥𝑛1𝑈𝑛1𝑥2,𝑝𝑛(𝑥)=1𝑥1±𝑥𝑖𝑛𝑃𝑛+1±𝑥𝑖2+±𝑥𝑖𝑛1𝑃𝑛±𝑥𝑖2,𝑝𝑛(𝑥)=1𝑥1±𝑥𝑖𝑛𝐹𝑛+1±+±𝑥𝑖𝑥𝑖𝑛1𝐹𝑛±,𝑝𝑥𝑖𝑛(𝑥)=1𝑥1±𝑥𝑛𝐵𝑛+±𝑥2𝑥𝑛1𝐵𝑛1,𝑝𝑥2𝑛(𝑥)=1𝑥1±𝑥2𝑛Φ𝑛+1±+±2𝑥𝑖𝑥2𝑛1Φ𝑛±,𝑝2𝑥𝑖𝑛(𝑥)=41𝑥1±𝑥𝑎𝑛𝐷𝑛±±𝑎𝑥𝑖,𝑎+4𝑥𝑎𝑛1𝐷𝑛1±.𝑎𝑥𝑖,𝑎(3.12) Since 𝑈𝑛+1(𝑦)=2𝑦𝑈𝑛(𝑦)𝑈𝑛1(𝑦), we have 𝑈𝑛+2(𝑦)=2𝑦𝑈𝑛+1(𝑦)𝑈𝑛(𝑦)=2𝑦2𝑦𝑈𝑛(𝑦)𝑈𝑛1(𝑦)𝑈𝑛=(𝑦)4𝑦2𝑈1𝑛(𝑦)2𝑦𝑈𝑛1(𝑦).(3.13) Hence, from the last expression of 𝑈𝑛+2 and the transfer formula of 𝑝𝑛(𝑥) in terms of 𝑈𝑛(𝑥) shown above, we obtain 𝑝𝑛(𝑥)=(±1)𝑛𝑥(𝑛2)/2𝑈𝑛+2𝑥2,(3.14) in which the case of 𝑝𝑛(𝑥)=(1)𝑛𝑥(𝑛2)/2𝑈𝑛+2𝑥2(3.15) was established in [24] using mathematical induction.
Equaling the right-hand expressions of the polynomials shown in each example, one may obtain various identities of generalized Gegenbauer-Humbert polynomials. For instance, from Example 2.3, we have 𝑈𝑛(𝑥)𝑥𝑈𝑛1(𝑥)=(1)𝑛𝑈𝑛(𝑥)+𝑥𝑈𝑛1(𝑥)=(±𝑖)𝑛𝑃𝑛+1(𝑥𝑖)𝑥(±𝑖)𝑛1𝑃𝑛(𝑥𝑖)=(±𝑖)𝑛𝐹𝑛+1(2𝑥𝑖)𝑥(±𝑖)𝑛1𝐹𝑛(2𝑥𝑖)=(±1)𝑛𝐵𝑛(±2𝑥2)(±1)𝑛1𝑥𝐵𝑛1=±1(±2𝑥2)2𝑛Φ𝑛+12±12𝑥𝑥2𝑛1Φ𝑛2=±12𝑥4𝑎𝑛𝐷𝑛2±1𝑎𝑥,𝑎𝑥4𝑎𝑛1𝐷𝑛12.𝑎𝑥,𝑎(3.16)

Using the relationship established in Theorem 2.1 and Corollaries 2.22.6, we may obtain some identities of polynomial sequences from the generalized Gegenbauer-Humbert polynomial sequence identity described in [5] 𝑃𝑛1,𝑦,𝐶(𝑥)=𝛼(𝑥)𝑃1,𝑦,𝐶𝑛1(𝑥)+𝐶2(2𝑥𝛼(𝑥)𝐶)(𝛽(𝑥))𝑛1,(3.17) where 𝑃𝑛1,𝑦,𝐶(𝑥) satisfies the recurrence relation of order 2, 𝑃𝑛1,𝑦,𝐶(𝑥)=𝑝(𝑥)𝑃1,𝑦,𝐶𝑛1(𝑥)+𝑞(𝑥)𝑃1,𝑦,𝐶𝑛2(𝑥) with coefficients 𝑝(𝑥) and 𝑞(𝑥), and 𝛼(𝑥)+𝛽(𝑥)=𝑝(𝑥) and 𝛼(𝑥)𝛽(𝑥)=𝑞(𝑥). Clearly (see (2.41) in [5]), 1𝛼(𝑥)=𝐶𝑥+𝑥2,1𝐶𝑦𝛽(𝑥)=𝐶𝑥𝑥2.𝐶𝑦(3.18) For 𝑦=𝐶=1, we have 𝑃𝑛1,1,1(𝑥)=𝑈𝑛(𝑥), where 𝑈𝑛(𝑥) are the Chebyshev polynomials of the second kind, and we can write (3.17) as 𝑈𝑛(𝑥)=𝛼(𝑥)𝑈𝑛1(𝑥)+(2𝑥𝛼(𝑥))(𝛽(𝑥))𝑛1=𝛼(𝑥)𝑈𝑛1(𝑥)+(𝛽(𝑥))𝑛,(3.19) where 𝛼(𝑥)=𝑥+𝑥21 and 𝛽(𝑥)=𝑥𝑥21. From the first formula of Example 3.2 and using transform ±1/(22𝑥)𝑥, we have 𝑈𝑛(𝑥)=(2𝑥)𝑛𝐽𝑛18𝑥2.(3.20) Substituting the above expression to (3.19) yields the identity (2𝑥)𝑛𝐽𝑛18𝑥2=𝑥+𝑥21(2𝑥)𝑛1𝐽𝑛118𝑥2+𝑥𝑥21𝑛.(3.21) Similarly, from Example 3.3, we obtain identities 𝑆𝑛(±2𝑥)=±𝑥+𝑥2𝑆1𝑛1(±2𝑥)+±𝑥𝑥21𝑛,(2𝑥±1)𝑛𝐻𝑛11±2𝑥=(2𝑥±1)𝑛1𝑥+𝑥2𝐻1𝑛11+1±2𝑥𝑥𝑥21𝑛.(3.22)

One may also extend some well-known identities of a polynomial sequence to other polynomial sequences using the relationships we have established. For instance, from the Cassini-like formula for Fibonacci polynomials 𝐹𝑛+1(𝑥)𝐹𝑛1(𝑥)𝐹2𝑛(𝑥)=(1)𝑛,(3.23) we use the relationship shown in Example 3.2 to obtain the Cassini-like formula for the Jacobsthal polynomials 𝐽𝑛(𝑥)𝐽𝑛2(𝑥)𝐽2𝑛1(𝑥)=(2𝑥)𝑛,(3.24) which can be transferred to the formula of Theorem  2 in [21] using the same argument in Example 3.2.

Similarly, from the transform 𝐹𝑛+1(𝑥)=(±𝑖)𝑛𝑈𝑛𝑥𝑖2,(3.25) we have 𝑈𝑛𝑥𝑖2𝑈𝑛2𝑥𝑖2𝑈2𝑛1𝑥𝑖2=(1)𝑛.(3.26)

To construct a transform relationship for the polynomials defined by recurrence relation with coefficients related to the order of polynomials is much more difficulty. One special example can be found on [25, page 240] by Andrews et al.. It seems there is no a general method applied to such polynomial sequences.

Acknowledgments

The authors wish to thank the referees and editors for their helpful comments and suggestions.

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