International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 268096 | 18 pages | https://doi.org/10.5402/2011/268096

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

Academic Editor: L. Vinet
Received19 Apr 2011
Accepted12 May 2011
Published30 Jun 2011

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 {π‘Žπ‘›(π‘₯)} is called a sequence of order 2 if it satisfies the linear recurrence relation of order 2π‘Žπ‘›(π‘₯)=𝑝(π‘₯)π‘Žπ‘›βˆ’1+π‘ž(π‘₯)π‘Žπ‘›βˆ’2(π‘₯),𝑛β‰₯2,(1.1) for some coefficient 𝑝(π‘₯)β‰’0 and π‘ž(π‘₯)β‰’0 and initial conditions π‘Ž0(π‘₯) and π‘Ž1(π‘₯). 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 {π‘Žπ‘›(π‘₯)} generated by (1.1). For the sake of the reader's convenience, we cite this result as follows.

Proposition 1.1. Let {π‘Žπ‘›(π‘₯)} be a sequence of order 2 satisfying the linear recurrence relation (1.1), then π‘Žπ‘›βŽ§βŽͺ⎨βŽͺβŽ©ξ‚΅π‘Ž(π‘₯)=1(π‘₯)βˆ’π›½(π‘₯)π‘Ž0(π‘₯)𝛼𝛼(π‘₯)βˆ’π›½(π‘₯)π‘›ξ‚΅π‘Ž(π‘₯)βˆ’1(π‘₯)βˆ’π›Ό(π‘₯)π‘Ž0(π‘₯)𝛽𝛼(π‘₯)βˆ’π›½(π‘₯)𝑛(π‘₯)if𝛼(π‘₯)≠𝛽(π‘₯),π‘›π‘Ž1(π‘₯)π›Όπ‘›βˆ’1(π‘₯)βˆ’(π‘›βˆ’1)π‘Ž0(π‘₯)𝛼𝑛(π‘₯)if𝛼(π‘₯)=𝛽(π‘₯),(1.2) where 𝛼(π‘₯) and 𝛽(π‘₯) are roots of 𝑑2βˆ’π‘(π‘₯)π‘‘βˆ’π‘ž(π‘₯)=0, namely, 1𝛼(π‘₯)=2ξ‚€βˆšπ‘(π‘₯)+𝑝21(π‘₯)+4π‘ž(π‘₯),𝛽(π‘₯)=2ξ‚€βˆšπ‘(π‘₯)βˆ’π‘2.(π‘₯)+4π‘ž(π‘₯)(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 π‘Ž0=0 and π‘Ž1=1, called the primary solution, can be expressed in terms of Chebyshev polynomial values. For instance, the authors show 𝐹𝑛=π‘–βˆ’π‘›π‘ˆπ‘›(𝑖/2)and 𝐿𝑛=2π‘–βˆ’π‘›π‘‡π‘›(𝑖/2), where 𝐹𝑛 and 𝐿𝑛 are, respectively, Fibonacci numbers and Lucas numbers, and 𝑇𝑛(π‘₯) and π‘ˆπ‘›(π‘₯) 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 {π‘Žπ‘›(π‘₯)} 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 π‘Ž0 and π‘Ž1, 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 {π‘ƒπ‘›πœ†,𝑦,𝐢(π‘₯)}𝑛β‰₯0 is defined by the expansion (see, e.g., [1], Gould [10], and the authors with Hsu [11]) ξ€·Ξ¦(𝑑)β‰‘πΆβˆ’2π‘₯𝑑+𝑦𝑑2ξ€Έβˆ’πœ†=𝑛β‰₯0π‘ƒπ‘›πœ†,𝑦,𝐢(π‘₯)𝑑𝑛,(1.4) where πœ†>0, 𝑦 and 𝐢≠0 are real numbers. As special cases of (1.4), we consider π‘ƒπ‘›πœ†,𝑦,𝐢(π‘₯) as follows (see [11]): 𝑃𝑛1,1,1(π‘₯)=π‘ˆπ‘›(π‘₯),Chebyshev polynomial of the second kind, 𝑃𝑛1/2,1,1(π‘₯)=πœ“π‘›(π‘₯), Legendre polynomial, 𝑃𝑛1,βˆ’1,1(π‘₯)=𝑃𝑛+1(π‘₯), Pell polynomial,𝑃𝑛1,βˆ’1,1(π‘₯/2)=𝐹𝑛+1(π‘₯), Fibonacci polynomial,𝑃𝑛1,1,1((π‘₯/2)+1)=𝐡𝑛(π‘₯), Morgan-Voyc polynomial, [12] by Koshy, 𝑃𝑛1,2,1(π‘₯/2)=Φ𝑛+1(π‘₯), Fermat polynomial of the first kind,𝑃𝑛1,2π‘Ž,2(π‘₯)=𝐷𝑛(π‘₯,π‘Ž), Dickson polynomial of the second kind,π‘Žβ‰ 0 (see, e.g., [13]) by Lidl et al.,

where π‘Ž is a real parameter, and 𝐹𝑛=𝐹𝑛(1) is the Fibonacci number. In particular, if 𝑦=𝐢=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 π‘ƒπ‘›πœ†,𝑦,𝐢(π‘₯)=2π‘₯πœ†+π‘›βˆ’1π‘ƒπΆπ‘›πœ†,𝑦,πΆπ‘›βˆ’1(π‘₯)βˆ’π‘¦2πœ†+π‘›βˆ’2π‘ƒπΆπ‘›πœ†,𝑦,πΆπ‘›βˆ’2(π‘₯),(1.5) for all 𝑛β‰₯2 with initial conditions 𝑃0πœ†,𝑦,𝐢(π‘₯)=Ξ¦(0)=πΆβˆ’πœ†,𝑃1πœ†,𝑦,𝐢(π‘₯)=Ξ¦ξ…ž(0)=2πœ†π‘₯πΆβˆ’πœ†βˆ’1,(1.6) the following theorem is obtained.

Theorem 1.2 (see [5]). Let √π‘₯≠±𝐢𝑦. The generalized Gegenbauer-Humbert polynomials {𝑃𝑛1,𝑦,𝐢(π‘₯)}𝑛β‰₯0 defined by expansion (1.4) can be expressed as 𝑃𝑛1,𝑦,𝐢(π‘₯)=πΆβˆ’π‘›βˆ’2ξ‚€π‘₯+√π‘₯2ξ‚βˆ’πΆπ‘¦π‘›+1βˆ’ξ‚€π‘₯βˆ’βˆšπ‘₯2ξ‚βˆ’πΆπ‘¦π‘›+12√π‘₯2.βˆ’πΆπ‘¦(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(π‘₯)≠𝛽(π‘₯), the first formula in (1.2) can be written as π‘Žπ‘›(π‘Žπ‘₯)=1(π‘₯)((𝛼(π‘₯))π‘›βˆ’(𝛽(π‘₯))𝑛)βˆ’π‘Ž0ξ€·(π‘₯)𝛼(π‘₯)𝛽(π‘₯)(𝛼(π‘₯))π‘›βˆ’1βˆ’(𝛽(π‘₯))π‘›βˆ’1ξ€Έ.𝛼(π‘₯)βˆ’π›½(π‘₯)(2.1) Noting that βˆ’π›Ό(π‘₯)𝛽(π‘₯)=𝛼(π‘₯)(𝛼(π‘₯)βˆ’π‘(π‘₯))=𝛽(π‘₯)(𝛽(π‘₯)βˆ’π‘(π‘₯)), we may further write the above expression of π‘Žπ‘›(π‘₯) as π‘Žπ‘›1(π‘₯)=π›Όξ€Ίπ‘Ž(π‘₯)βˆ’π›½(π‘₯)1(π‘₯)((𝛼(π‘₯))π‘›βˆ’(𝛽(π‘₯))𝑛)+π‘Ž0(π‘₯)𝛼(π‘₯)(𝛼(π‘₯)βˆ’π‘(π‘₯))Γ—(𝛼(π‘₯))π‘›βˆ’1βˆ’π‘Ž0(π‘₯)𝛽(π‘₯)(𝛽(π‘₯)βˆ’π‘(π‘₯))(𝛽(π‘₯))π‘›βˆ’1ξ€»=π‘Ž0ξ€·(π‘₯)(𝛼(π‘₯))𝑛+1βˆ’(𝛽(π‘₯))𝑛+1ξ€Έ+ξ€·π‘Ž1(π‘₯)βˆ’π‘Ž0ξ€Έ(π‘₯)𝑝(π‘₯)((𝛼(π‘₯))π‘›βˆ’(𝛽(π‘₯))𝑛)𝛼.(π‘₯)βˆ’π›½(π‘₯)(2.2)

Denote π‘Ÿ(π‘₯)=π‘₯+√π‘₯2βˆ’πΆπ‘¦ and 𝑠(π‘₯)=π‘₯βˆ’βˆšπ‘₯2βˆ’πΆπ‘¦. To find a transfer formula between expressions (1.7) and (2.2), we set 𝛼(π‘₯)∢=π‘Ÿ(π‘₯)π‘˜(π‘₯),𝛽(π‘₯)∢=𝑠(π‘₯)π‘˜(π‘₯),(2.3) for a nonzero real or complex-valued function π‘˜(π‘₯), which are two roots of 𝑑2βˆ’π‘(π‘₯)π‘‘βˆ’π‘ž(π‘₯)=0. Thus, adding and multiplying two equations of (2.3) side by side, we obtain 𝛼(π‘₯)+𝛽(π‘₯)=𝑝(π‘₯)=2π‘₯π‘˜,(π‘₯)𝛼(π‘₯)𝛽(π‘₯)=βˆ’π‘ž(π‘₯)=𝐢𝑦(π‘˜(π‘₯))2.(2.4) The above system implies ξ‚™π‘˜(π‘₯)=±𝐢𝑦,βˆ’π‘ž(π‘₯)(2.5) and at π‘₯=𝑝(π‘₯)π‘˜(π‘₯)2=±𝑝(π‘₯)2𝐢𝑦,βˆ’π‘ž(π‘₯)(2.6)π‘Ÿ(π‘₯) and 𝑠(π‘₯) give expressions of 𝛼(π‘₯) and 𝛽(π‘₯) as π‘Ÿξ‚€βˆšπ›Ό(π‘₯)=Β±(𝑝(π‘₯)/2)𝐢𝑦/βˆ’π‘ž(π‘₯)Β±βˆšπ‘ ξ‚€βˆšπΆπ‘¦/βˆ’π‘ž(π‘₯),𝛽(π‘₯)=Β±(𝑝(π‘₯)/2)𝐢𝑦/βˆ’π‘ž(π‘₯)Β±βˆšπΆπ‘¦/βˆ’π‘ž(π‘₯).(2.7) It is clear that 𝛼(π‘₯) and 𝛽(π‘₯) satisfy 𝛼(π‘₯)+𝛽(π‘₯)=𝑝(π‘₯) and 𝛼(π‘₯)𝛽(π‘₯)=βˆ’π‘ž(π‘₯).

We first consider the case of βˆšπ‘˜(π‘₯)=βˆ’πΆπ‘¦/π‘ž(π‘₯). Substituting the corresponding (2.7) with positive sign into (2.2), we have π‘Žπ‘›π‘Ž(π‘₯)=0ξ€·π‘Ÿ(π‘₯)𝑛+1(π‘₯)βˆ’π‘ π‘›+1ξ€Έξ€·π‘Ž(π‘₯)+π‘˜(π‘₯)1(π‘₯)βˆ’π‘Ž0ξ€Έ(π‘₯)𝑝(π‘₯)(π‘Ÿπ‘›(π‘₯)βˆ’π‘ π‘›(π‘₯))π‘˜π‘›(π‘₯)(π‘Ÿ(π‘₯)βˆ’π‘ (π‘₯))=π‘Ž0(π‘₯)𝐢𝑛+2βŽ›βŽœβŽœβŽξƒŽβˆ’π‘ž(π‘₯)βŽžβŽŸβŽŸβŽ πΆπ‘¦π‘›π‘ƒπ‘›1,𝑦,πΆξ‚΅π‘˜(π‘₯)𝑝(π‘₯)2ξ‚Ά+ξ€·π‘Ž1(π‘₯)βˆ’π‘Ž0𝐢(π‘₯)𝑝(π‘₯)𝑛+1βŽ›βŽœβŽœβŽξƒŽβˆ’π‘ž(π‘₯)βŽžβŽŸβŽŸβŽ πΆπ‘¦π‘›βˆ’1𝑃1,𝑦,πΆπ‘›βˆ’1ξ‚΅π‘˜(π‘₯)𝑝(π‘₯)2ξ‚Ά=π‘Ž0(π‘₯)𝐢𝑛+2βŽ›βŽœβŽœβŽξƒŽβˆ’π‘ž(π‘₯)βŽžβŽŸβŽŸβŽ πΆπ‘¦π‘›π‘ƒπ‘›1,𝑦,𝐢𝑝(π‘₯)2𝐢𝑦ξƒͺ+ξ€·π‘Žβˆ’π‘ž(π‘₯)1(π‘₯)βˆ’π‘Ž0𝐢(π‘₯)𝑝(π‘₯)𝑛+1βŽ›βŽœβŽœβŽξƒŽβˆ’π‘ž(𝑋)βŽžβŽŸβŽŸβŽ πΆπ‘¦π‘›βˆ’1𝑃1,𝑦,πΆπ‘›βˆ’1𝑝(π‘₯)2𝐢𝑦ξƒͺ.βˆ’π‘ž(π‘₯)(2.8) Similarly, for βˆšπ‘˜(π‘₯)=βˆ’βˆ’πΆπ‘¦/π‘ž(π‘₯), we have π‘Žπ‘›(π‘₯)=π‘Ž0(π‘₯)𝐢𝑛+2βŽ›βŽœβŽœβŽβˆ’ξƒŽβˆ’π‘ž(π‘₯)βŽžβŽŸβŽŸβŽ πΆπ‘¦π‘›π‘ƒπ‘›1,𝑦,πΆξƒ©βˆ’π‘(π‘₯)2𝐢𝑦ξƒͺ+ξ€·π‘Žβˆ’π‘ž(π‘₯)1(π‘₯)βˆ’π‘Ž0𝐢(π‘₯)𝑝(π‘₯)𝑛+1βŽ›βŽœβŽœβŽβˆ’ξƒŽβˆ’π‘ž(π‘₯)βŽžβŽŸβŽŸβŽ πΆπ‘¦π‘›βˆ’1𝑃1,𝑦,πΆπ‘›βˆ’1ξƒ©βˆ’π‘(π‘₯)2𝐢𝑦ξƒͺ.βˆ’π‘ž(π‘₯)(2.9)

Therefore, we obtain our main result.

Theorem 2.1. Let sequence {π‘Žπ‘›(π‘₯)}𝑛β‰₯0 be defined by π‘Žπ‘›(π‘₯)=𝑝(π‘₯)π‘Žπ‘›βˆ’1(π‘₯)+π‘ž(π‘₯)π‘Žπ‘›βˆ’2(π‘₯)(𝑛β‰₯2) with initial conditions π‘Ž0(π‘₯) and π‘Ž1(π‘₯), then π‘Žπ‘›(π‘₯) can be presented as π‘Žπ‘›(π‘₯)=π‘Ž0(π‘₯)𝐢𝑛+2βŽ›βŽœβŽœβŽΒ±ξƒŽβˆ’π‘ž(π‘₯)βŽžβŽŸβŽŸβŽ πΆπ‘¦π‘›π‘ƒπ‘›1,𝑦,𝐢±𝑝(π‘₯)2𝐢𝑦ξƒͺ+ξ€·π‘Žβˆ’π‘ž(π‘₯)1(π‘₯)βˆ’π‘Ž0𝐢(π‘₯)𝑝(π‘₯)𝑛+1βŽ›βŽœβŽœβŽΒ±ξƒŽβˆ’π‘ž(𝑋)βŽžβŽŸβŽŸβŽ πΆπ‘¦π‘›βˆ’1𝑃1,𝑦,πΆπ‘›βˆ’1±𝑝(π‘₯)2𝐢𝑦ξƒͺ,βˆ’π‘ž(π‘₯)(2.10) where {𝑃𝑛1,𝑦,𝑐} is the sequence of any generalized Gegenbauer-Humbert polynomials with πœ†=1. In particular, π‘Žπ‘›(π‘₯) can be expressed in terms of {𝑃𝑛1,1,1=π‘ˆπ‘›}, the sequence of the Chebyshev polynomials of the second kind, π‘Žπ‘›(π‘₯)=π‘Ž0ξ‚€Β±βˆš(π‘₯)ξ‚βˆ’π‘ž(π‘₯)π‘›π‘ˆπ‘›ξƒ©Β±π‘(π‘₯)2√ξƒͺ+ξ€·π‘Žβˆ’π‘ž(π‘₯)1(π‘₯)βˆ’π‘Ž0ξ€Έξ‚€Β±βˆš(π‘₯)𝑝(π‘₯)ξ‚βˆ’π‘ž(π‘₯)π‘›βˆ’1π‘ˆπ‘›βˆ’1±𝑝(π‘₯)2√ξƒͺ,βˆ’π‘ž(π‘₯)(2.11) which is a special case of (2.10) for (𝑦,𝐢)=(1,1).

Corollary 2.2. For (𝑦,𝐢)=(βˆ’1,1),(1,1),(2,1),π‘Žπ‘›π‘‘(2π‘Ž,2)(π‘Žβ‰ 0), respectively, from (2.10), one has transfer formulas π‘Žπ‘›(π‘₯)=π‘Ž0ξ‚€Β±βˆš(π‘₯)ξ‚π‘ž(π‘₯)𝑛𝑃𝑛+1±𝑝(π‘₯)2√ξƒͺ+ξ€·π‘Žπ‘ž(π‘₯)1(π‘₯)βˆ’π‘Ž0ξ€Έξ‚€Β±βˆš(π‘₯)𝑝(π‘₯)ξ‚π‘ž(π‘₯)π‘›βˆ’1𝑃𝑛±𝑝(π‘₯)2√ξƒͺ,π‘Žπ‘ž(π‘₯)𝑛(π‘₯)=π‘Ž0ξ‚€Β±βˆš(π‘₯)π‘žξ‚(π‘₯)𝑛𝐹𝑛+1±𝑝(π‘₯)βˆšπ‘žξƒͺ+ξ€·π‘Ž(π‘₯)1(π‘₯)βˆ’π‘Ž0ξ€Έξ‚€Β±βˆš(π‘₯)𝑝(π‘₯)ξ‚π‘ž(π‘₯)π‘›βˆ’1𝐹𝑛±𝑝(π‘₯)√ξƒͺ,π‘Žπ‘ž(π‘₯)𝑛(π‘₯)=π‘Ž0ξ‚€Β±βˆš(π‘₯)ξ‚βˆ’π‘ž(π‘₯)𝑛𝐡𝑛±𝑝(π‘₯)√ξƒͺ+ξ€·π‘Žβˆ’π‘ž(π‘₯)βˆ’21(π‘₯)βˆ’π‘Ž0ξ€Έξ‚€Β±βˆš(π‘₯)𝑝(π‘₯)ξ‚βˆ’π‘ž(π‘₯)π‘›βˆ’1π΅π‘›βˆ’1±𝑝(π‘₯)√ξƒͺ,π‘Žβˆ’π‘ž(π‘₯)βˆ’2𝑛(π‘₯)=π‘Ž0±(π‘₯)βˆ’π‘ž(π‘₯)2ξƒͺ𝑛Φ𝑛+1±𝑝(π‘₯)2ξ‚Ά+ξ€·π‘Žβˆ’π‘ž(π‘₯)1(π‘₯)βˆ’π‘Ž0±(π‘₯)𝑝(π‘₯)βˆ’π‘ž(π‘₯)2ξƒͺπ‘›βˆ’1Φ𝑛±𝑝(π‘₯)2ξ‚Ά,π‘Žβˆ’π‘ž(π‘₯)𝑛(π‘₯)=4π‘Ž0±(π‘₯)βˆ’π‘ž(π‘₯)π‘Žξƒͺ𝑛𝐷𝑛±𝑝(π‘₯)π‘Žξ‚Άξ€·π‘Žβˆ’π‘ž(π‘₯),π‘Ž+41(π‘₯)βˆ’π‘Ž0±(π‘₯)𝑝(π‘₯)βˆ’π‘ž(π‘₯)π‘Žξƒͺπ‘›βˆ’1π·π‘›βˆ’1±𝑝(π‘₯)π‘Žξ‚Ά,βˆ’π‘ž(π‘₯),π‘Ž(2.12) where π‘ˆπ‘›(π‘₯), 𝑃𝑛(π‘₯), 𝐹𝑛(π‘₯), Φ𝑛(π‘₯), and 𝐷𝑛(π‘₯,π‘Ž) 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(π‘₯)=cos(𝑛arccosπ‘₯) satisfying recurrence relation (1.1) with 𝑝(π‘₯)=2π‘₯ and π‘ž=βˆ’1 and initial conditions 𝑇0(π‘₯)=1 and 𝑇1(π‘₯)=π‘₯. From Corollary 2.2, we have 𝑇𝑛(π‘₯)=π‘ˆπ‘›(π‘₯)βˆ’π‘₯π‘ˆπ‘›βˆ’1𝑇(π‘₯),𝑛(π‘₯)=(βˆ’1)π‘›ξ€·π‘ˆπ‘›(βˆ’π‘₯)+π‘₯π‘ˆπ‘›βˆ’1ξ€Έ,𝑇(π‘₯)𝑛(π‘₯)=(±𝑖)𝑛𝑃𝑛+1(βˆ“π‘₯𝑖)βˆ’π‘₯(±𝑖)π‘›βˆ’1𝑃𝑛𝑇(βˆ“π‘₯𝑖),𝑛(π‘₯)=(±𝑖)𝑛𝐹𝑛+1(βˆ“2π‘₯𝑖)βˆ’π‘₯(±𝑖)π‘›βˆ’1𝐹𝑛𝑇(βˆ“2π‘₯𝑖),𝑛(π‘₯)=(Β±1)𝑛𝐡𝑛(Β±2π‘₯βˆ’2)βˆ’(Β±1)π‘›βˆ’1π‘₯π΅π‘›βˆ’1𝑇(Β±2π‘₯βˆ’2),𝑛(π‘₯)=𝐡𝑛(Β±2π‘₯βˆ’2)βˆ’π‘₯π΅π‘›βˆ’1(𝑇±2π‘₯βˆ’2),𝑛±1(π‘₯)=√2ξƒͺ𝑛Φ𝑛+1ξ‚€βˆšβˆ“2±12π‘₯βˆ’π‘₯√2ξƒͺπ‘›βˆ’1Ξ¦π‘›ξ‚€βˆšβˆ“2,𝑇2π‘₯𝑛±1(π‘₯)=√ξƒͺ4π‘Žπ‘›π·π‘›ξ‚€βˆšβˆ“2±1π‘Žπ‘₯,π‘Žβˆ’π‘₯√ξƒͺ4π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξ‚€βˆšβˆ“2,π‘Žπ‘₯,π‘Ž(2.13) in which the first relation is equivalent to the well-known result 2𝑇𝑛(π‘₯)=π‘ˆπ‘›(π‘₯)βˆ’π‘ˆπ‘›βˆ’2(π‘₯) due to 2𝑇𝑛(π‘₯)=2π‘ˆπ‘›(π‘₯)βˆ’2π‘₯π‘ˆπ‘›βˆ’1(π‘₯)=π‘ˆπ‘›ξ€·(π‘₯)+2π‘₯π‘ˆπ‘›βˆ’1(π‘₯)βˆ’π‘ˆπ‘›βˆ’2ξ€Έ(π‘₯)βˆ’2π‘₯π‘ˆπ‘›βˆ’1(π‘₯).(2.14)
For the special cases of π‘Ž0(π‘₯) and π‘Ž1(π‘₯), we have the following corollaries.

Corollary 2.4. Let sequence {π‘Žπ‘›(π‘₯)}𝑛β‰₯0 be defined by π‘Žπ‘›(π‘₯)=𝑝(π‘₯)π‘Žπ‘›βˆ’1(π‘₯)+π‘ž(π‘₯)π‘Žπ‘›βˆ’2(π‘₯) (𝑛β‰₯2) with initial conditions π‘Ž0(π‘₯)=0 and π‘Ž1(π‘₯)=𝑑. Then π‘Žπ‘›ξ‚€Β±βˆš(π‘₯)=π‘‘ξ‚βˆ’π‘ž(π‘₯)π‘›βˆ’1π‘ˆπ‘›βˆ’1±𝑝(π‘₯)2√ξƒͺ,π‘Žβˆ’π‘ž(π‘₯)π‘›ξ‚€Β±βˆš(π‘₯)=π‘‘ξ‚π‘ž(π‘₯)π‘›βˆ’1𝑃𝑛±𝑝(π‘₯)2√ξƒͺ,π‘Žπ‘ž(π‘₯)π‘›ξ‚€Β±βˆš(π‘₯)=π‘‘π‘žξ‚(π‘₯)π‘›βˆ’1𝐹𝑛±𝑝(π‘₯)βˆšπ‘žξƒͺ,π‘Ž(π‘₯)π‘›ξ‚€Β±βˆš(π‘₯)=π‘‘ξ‚βˆ’π‘ž(π‘₯)π‘›βˆ’1π΅π‘›βˆ’1±𝑝(π‘₯)√ξƒͺ,π‘Žβˆ’π‘ž(π‘₯)βˆ’2𝑛±(π‘₯)=π‘‘βˆ’π‘ž(π‘₯)2ξƒͺπ‘›βˆ’1Φ𝑛±𝑝(π‘₯)2ξ‚Ά,π‘Žβˆ’π‘ž(π‘₯)𝑛±(π‘₯)=4π‘‘βˆ’π‘ž(π‘₯)π‘Žξƒͺπ‘›βˆ’1π·π‘›βˆ’1±𝑝(π‘₯)π‘Žξ‚Ά.βˆ’π‘ž(π‘₯),π‘Ž(2.15)

Corollary 2.5. Let sequence {π‘Žπ‘›(π‘₯)}𝑛β‰₯0 be defined by π‘Žπ‘›(π‘₯)=𝑝(π‘₯)π‘Žπ‘›βˆ’1(π‘₯)+π‘ž(π‘₯)π‘Žπ‘›βˆ’2(π‘₯)(𝑛β‰₯2) with initial conditions π‘Ž0(π‘₯)=𝑐 and π‘Ž1(π‘₯)=𝑐𝑝(π‘₯), then π‘Žπ‘›ξ‚€Β±βˆš(π‘₯)=π‘ξ‚βˆ’π‘ž(π‘₯)π‘›π‘ˆπ‘›ξƒ©Β±π‘(π‘₯)2√ξƒͺ,π‘Žβˆ’π‘ž(π‘₯)π‘›ξ‚€Β±βˆš(π‘₯)=π‘ξ‚π‘ž(π‘₯)𝑛𝑃𝑛+1±𝑝(π‘₯)2√ξƒͺ,π‘Žπ‘ž(π‘₯)π‘›ξ‚€Β±βˆš(π‘₯)=π‘π‘žξ‚(π‘₯)𝑛𝐹𝑛+1±𝑝(π‘₯)βˆšπ‘žξƒͺ,π‘Ž(π‘₯)π‘›ξ‚€Β±βˆš(π‘₯)=π‘ξ‚βˆ’π‘ž(π‘₯)𝑛𝐡𝑛±𝑝(π‘₯)√ξƒͺ,π‘Žβˆ’π‘ž(π‘₯)βˆ’2𝑛(±π‘₯)=π‘βˆ’π‘ž(π‘₯)2ξƒͺ𝑛Φ𝑛+1±𝑝(π‘₯)2ξ‚Ά,π‘Žβˆ’π‘ž(π‘₯)𝑛±(π‘₯)=4π‘βˆ’π‘ž(π‘₯)π‘Žξƒͺ𝑛𝐷𝑛±𝑝(π‘₯)π‘Žξ‚Ά.βˆ’π‘ž(π‘₯),π‘Ž(2.16)

We now give another special case of Theorem 2.1 for the sequence defined by (1.1) with initial cases π‘Ž0(π‘₯)=2 and π‘Ž1(π‘₯)=𝑝(π‘₯).

Corollary 2.6. Let sequence {π‘Žπ‘›(π‘₯)}𝑛β‰₯0 be defined by π‘Žπ‘›(π‘₯)=𝑝(π‘₯)π‘Žπ‘›βˆ’1(π‘₯)+π‘ž(π‘₯)π‘Žπ‘›βˆ’2(π‘₯)(𝑛β‰₯2) with initial conditions π‘Ž0(π‘₯)=2 and π‘Ž1(π‘₯)=𝑝(π‘₯).
Then π‘Žπ‘›ξ‚€Β±βˆš(π‘₯)=2ξ‚βˆ’π‘ž(π‘₯)π‘›π‘ˆπ‘›ξƒ©Β±π‘(π‘₯)2√ξƒͺξ‚€Β±βˆšβˆ’π‘ž(π‘₯)βˆ’π‘(π‘₯)ξ‚βˆ’π‘ž(π‘₯)π‘›βˆ’1π‘ˆπ‘›βˆ’1±𝑝(π‘₯)2√ξƒͺ,π‘Žβˆ’π‘ž(π‘₯)π‘›ξ‚€Β±βˆš(π‘₯)=2π‘žξ‚(π‘₯)𝑛𝑃𝑛+1±𝑝(π‘₯)2βˆšπ‘žξƒͺξ‚€Β±βˆš(π‘₯)βˆ’π‘(π‘₯)ξ‚π‘ž(π‘₯)π‘›βˆ’1𝑃𝑛±𝑝(π‘₯)2√ξƒͺ,π‘Žπ‘ž(π‘₯)π‘›ξ‚€Β±βˆš(π‘₯)=2ξ‚π‘ž(π‘₯)𝑛𝐹𝑛+1±𝑝(π‘₯)√ξƒͺξ‚€Β±βˆšπ‘ž(π‘₯)βˆ’π‘(π‘₯)ξ‚π‘ž(π‘₯)π‘›βˆ’1𝐹𝑛±𝑝(π‘₯)√ξƒͺ,π‘Žπ‘ž(π‘₯)π‘›ξ‚€Β±βˆš(π‘₯)=2ξ‚βˆ’π‘ž(π‘₯)𝑛𝐡𝑛±𝑝(π‘₯)√ξƒͺξ‚€Β±βˆšβˆ’π‘ž(π‘₯)βˆ’2βˆ’π‘(π‘₯)ξ‚βˆ’π‘ž(π‘₯)π‘›βˆ’1π΅π‘›βˆ’1±𝑝(π‘₯)√ξƒͺ,π‘Žβˆ’π‘ž(π‘₯)βˆ’2𝑛±(π‘₯)=2βˆ’π‘ž(π‘₯)2ξƒͺ𝑛Φ𝑛+1±𝑝(π‘₯)2ξ‚Άξƒ©Β±ξ‚™βˆ’π‘ž(π‘₯)βˆ’π‘(π‘₯)βˆ’π‘ž(π‘₯)2ξƒͺπ‘›βˆ’1Φ𝑛±𝑝(π‘₯)2ξ‚Ά,π‘Žβˆ’π‘ž(π‘₯)𝑛(π‘₯)=23ξƒ©Β±ξ‚™βˆ’π‘ž(π‘₯)π‘Žξƒͺ𝑛𝐷𝑛±𝑝(π‘₯)π‘Žξ‚Άβˆ’π‘ž(π‘₯),π‘Žβˆ’π‘(π‘₯)22ξƒ©Β±ξ‚™βˆ’π‘ž(π‘₯)π‘Žξƒͺπ‘›βˆ’1π·π‘›βˆ’1±𝑝(π‘₯)π‘Žξ‚Ά.βˆ’π‘ž(π‘₯),π‘Ž(2.17) In addition, one has π‘Žπ‘›ξ‚€Β±βˆš(π‘₯)=2ξ‚βˆ’π‘ž(π‘₯)𝑛𝑇𝑛±𝑝(π‘₯)2√ξƒͺβˆ’π‘ž(π‘₯),(2.18) where 𝑇𝑛(π‘₯) 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 π‘ˆπ‘›(π‘₯)=2π‘₯π‘ˆπ‘›βˆ’1(π‘₯)βˆ’π‘ˆπ‘›βˆ’2(π‘₯), one easily sees π‘Žπ‘›ξ‚€βˆš(π‘₯)=ξ‚βˆ’π‘ž(π‘₯)𝑛2π‘ˆπ‘›ξƒ©π‘(π‘₯)2√ξƒͺβˆ’βˆ’π‘ž(π‘₯)𝑝(π‘₯)βˆšπ‘ˆβˆ’π‘ž(π‘₯)π‘›βˆ’1𝑝(π‘₯)2√=ξ‚€βˆšβˆ’π‘ž(π‘₯)ξƒͺξƒ­ξ‚βˆ’π‘ž(π‘₯)𝑛2π‘ˆπ‘›ξƒ©π‘(π‘₯)2√ξƒͺβˆ’ξƒ©π‘ˆβˆ’π‘ž(π‘₯)𝑛𝑝(π‘₯)2√ξƒͺβˆ’π‘ž(π‘₯)+π‘ˆπ‘›βˆ’2𝑝(π‘₯)2√=ξ‚€βˆšβˆ’π‘ž(π‘₯)ξƒͺξƒͺξƒ­ξ‚βˆ’π‘ž(π‘₯)π‘›ξƒ¬π‘ˆπ‘›ξƒ©π‘(π‘₯)2√ξƒͺβˆ’π‘ž(π‘₯)βˆ’π‘ˆπ‘›βˆ’2𝑝(π‘₯)2√.βˆ’π‘ž(π‘₯)ξƒͺξƒ­(2.19) From the first formula of Example 2.3, the above last expression of π‘Žπ‘›(π‘₯) 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(π‘₯)} defined by (1.1) with 𝑝(π‘₯)=π‘₯ and π‘ž(π‘₯)=1 and initial conditions 𝐿0(π‘₯)=2 and 𝐿1(π‘₯)=π‘₯ has an explicit formula for its general term 𝐿𝑛(π‘₯)=2(±𝑖)π‘›π‘‡π‘›ξ‚€βˆ“π‘₯𝑖2.(2.20) Using Corollary 2.6, we also have 𝐿𝑛(π‘₯)=2(±𝑖)π‘›π‘ˆπ‘›ξ‚€βˆ“π‘₯𝑖2ξ‚βˆ’π‘₯(±𝑖)π‘›βˆ’1π‘ˆπ‘›βˆ’1ξ‚€βˆ“π‘₯𝑖2,𝐿𝑛(π‘₯)=2𝑃𝑛+1ξ‚€Β±π‘₯2ξ‚βˆ’π‘₯𝑃𝑛±π‘₯2,𝐿𝑛(π‘₯)=2𝐹𝑛+1(Β±π‘₯)βˆ’π‘₯𝐹𝑛𝐿(Β±π‘₯),𝑛(π‘₯)=2(±𝑖)𝑛𝐡𝑛(βˆ“π‘₯π‘–βˆ’2)βˆ’π‘₯(±𝑖)π‘›βˆ’1π΅π‘›βˆ’1𝐿(βˆ“π‘₯π‘–βˆ’2),𝑛±𝑖(π‘₯)=2√2ξƒͺ𝑛Φ𝑛+1ξ‚€βˆ“βˆšξ‚ξƒ©Β±π‘–2π‘₯π‘–βˆ’π‘₯√2ξƒͺπ‘›βˆ’1Ξ¦π‘›ξ‚€βˆ“βˆšξ‚,𝐿2π‘₯𝑖𝑛(π‘₯)=23ξƒ©Β±π‘–βˆšπ‘Žξƒͺπ‘›π·π‘›ξ‚€βˆ“βˆšξ‚π‘Žπ‘₯𝑖,π‘Žβˆ’π‘₯22ξƒ©Β±π‘–βˆšπ‘Žξƒͺπ‘›βˆ’1π·π‘›βˆ’1ξ‚€βˆ“βˆšξ‚.π‘Žπ‘₯𝑖,π‘Ž(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 {π‘Žπ‘›(π‘₯)} 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, 2π‘₯βˆ’1 and 2π‘₯+1, respectively (see, e.g., Mason and Handscomb [18] and Rivlin [19]). Hence, the Chebyshev polynomials of the third kind, 𝑇𝑛(3)(π‘₯), and the Chebyshev polynomials of the fourth kind, 𝑇𝑛(4)(π‘₯), when π‘₯2β‰ 1, have the following expressions using the argument shown in [5]: 𝑇𝑛(3)√(π‘₯)=π‘₯2βˆ’1+π‘₯βˆ’12√π‘₯2ξ‚€βˆšβˆ’1π‘₯+π‘₯2ξ‚βˆ’1𝑛+√π‘₯2βˆ’1βˆ’π‘₯+12√π‘₯2ξ‚€βˆšβˆ’1π‘₯βˆ’π‘₯2ξ‚βˆ’1𝑛,𝑇𝑛(4)√(π‘₯)=π‘₯2βˆ’1+π‘₯+12√π‘₯2ξ‚€βˆšβˆ’1π‘₯+π‘₯2ξ‚βˆ’1𝑛+√π‘₯2βˆ’1βˆ’π‘₯βˆ’12√π‘₯2ξ‚€βˆšβˆ’1π‘₯βˆ’π‘₯2ξ‚βˆ’1𝑛.(3.1)

Similarly to the Chebyshev polynomials of the first kind (see Example 2.3), we can transfer 𝑇𝑛(3)(π‘₯) and 𝑇𝑛(4)(π‘₯) to the generalized Gegenbauer-Humbert polynomials with πœ†=1, 𝑇𝑛(3)(π‘₯)=π‘ˆπ‘›(π‘₯)βˆ’π‘ˆπ‘›βˆ’1𝑇(π‘₯),𝑛(3)(π‘₯)=(βˆ’1)π‘›ξ€·π‘ˆπ‘›(βˆ’π‘₯)+π‘ˆπ‘›βˆ’1(ξ€Έ,𝑇π‘₯)𝑛(3)(π‘₯)=(±𝑖)𝑛𝑃𝑛+1(βˆ“π‘₯𝑖)βˆ’(±𝑖)π‘›βˆ’1𝑃𝑛𝑇(βˆ“π‘₯𝑖),𝑛(3)(π‘₯)=(±𝑖)𝑛𝐹𝑛+1(βˆ“2π‘₯𝑖)βˆ’(±𝑖)π‘›βˆ’1𝐹𝑛𝑇(βˆ“2π‘₯𝑖),𝑛(3)(π‘₯)=(Β±1)𝑛𝐡𝑛(Β±2π‘₯βˆ’2)βˆ’(Β±1)π‘›βˆ’1π΅π‘›βˆ’1𝑇(Β±2π‘₯βˆ’2),𝑛(3)±1(π‘₯)=√2ξƒͺ𝑛Φ𝑛+1ξ‚€βˆšβˆ“2ξ‚βˆ’ξƒ©Β±12π‘₯√2ξƒͺπ‘›βˆ’1Ξ¦π‘›ξ‚€βˆšβˆ“2,𝑇2π‘₯𝑛(3)±1(π‘₯)=√ξƒͺ4π‘Žπ‘›π·π‘›ξ‚€βˆšβˆ“2ξ‚βˆ’ξƒ©Β±1π‘Žπ‘₯,π‘Žβˆšξƒͺ4π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξ‚€βˆšβˆ“2,π‘‡π‘Žπ‘₯,π‘Žπ‘›(4)(π‘₯)=π‘ˆπ‘›(π‘₯)+π‘ˆπ‘›βˆ’1𝑇(π‘₯),𝑛(4)(π‘₯)=(βˆ’1)π‘›ξ€·π‘ˆπ‘›(βˆ’π‘₯)βˆ’π‘ˆπ‘›βˆ’1ξ€Έ,𝑇(π‘₯)𝑛(4)(π‘₯)=(±𝑖)𝑛𝑃𝑛+1(βˆ“π‘₯𝑖)+(±𝑖)π‘›βˆ’1𝑃𝑛𝑇(βˆ“π‘₯𝑖),𝑛(4)(π‘₯)=(±𝑖)𝑛𝐹𝑛+1(βˆ“2π‘₯𝑖)+(±𝑖)π‘›βˆ’1𝐹𝑛(π‘‡βˆ“2π‘₯𝑖),𝑛(4)(π‘₯)=(Β±1)𝑛𝐡𝑛(Β±2π‘₯βˆ’2)+(Β±1)π‘›βˆ’1π΅π‘›βˆ’1𝑇(Β±2π‘₯βˆ’2),𝑛(4)±1(π‘₯)=√2ξƒͺ𝑛Φ𝑛+1ξ‚€βˆšβˆ“2+±12π‘₯√2ξƒͺπ‘›βˆ’1Ξ¦π‘›ξ‚€βˆšβˆ“2,𝑇2π‘₯𝑛(3)±1(π‘₯)=√ξƒͺ4π‘Žπ‘›π·π‘›ξ‚€βˆšβˆ“2+±1π‘Žπ‘₯,π‘Žβˆšξƒͺ4π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξ‚€βˆšβˆ“2.π‘Žπ‘₯,π‘Ž(3.2) From the above formulas, one may obtain some identities between the Chebyshev polynomials of different kinds. For instance, 𝑇𝑛(3)(π‘₯)+𝑇𝑛(4)(π‘₯)=2π‘ˆπ‘›π‘‡(π‘₯),𝑛(π‘₯)+π‘₯𝑇𝑛(4)(π‘₯)=(1+π‘₯)π‘ˆπ‘›(𝑇π‘₯),𝑛(π‘₯)βˆ’π‘₯𝑇𝑛(3)(π‘₯)=(1βˆ’π‘₯)π‘ˆπ‘›(π‘₯).(3.3)

Since 𝑇𝑛(π‘₯)=cosπ‘›πœƒ, π‘ˆπ‘›(π‘₯)=sin(𝑛+1)πœƒ/sinπœƒ, 𝑇𝑛(3)(π‘₯)=cos(𝑛+1/2)πœƒ/cos(1/2)πœƒ, and 𝑇𝑛(4)(π‘₯)=sin(𝑛+1/2)πœƒ/sin(1/2)πœƒ, where π‘₯=cosπœƒ, the above identities of Chebyshev polynomials also present the following identities of trigonometric functions, respectively, cos(𝑛+1/2)πœƒ+cos(1/2)πœƒsin(𝑛+1/2)πœƒsin(1/2)πœƒ=2sin(𝑛+1)πœƒ,sinπœƒcosπ‘›πœƒ+cosπœƒsin(𝑛+1/2)πœƒ=sin(1/2)πœƒ(1+cosπœƒ)sin(𝑛+1)πœƒ,sinπœƒcosπ‘›πœƒβˆ’cosπœƒsin(𝑛+1/2)πœƒsin(1/2)πœƒ=(1βˆ’cosπœƒ)sin(𝑛+1)πœƒ.sinπœƒ(3.4)

Example 3.2. Consider the Jacobsthal polynomials {𝐽𝑛(π‘₯)} defined by (1.1) with coefficients 𝑝(π‘₯)=1 and π‘ž(π‘₯)=2π‘₯ and initial conditions 𝐽0(π‘₯)=𝐽1(π‘₯)=1. One may use Corollary 2.5 to obtain transfer formulas π½π‘›ξ‚€Β±βˆš(π‘₯)=ξ‚βˆ’2π‘₯π‘›π‘ˆπ‘›ξƒ©Β±12√ξƒͺ,π½βˆ’2π‘₯π‘›ξ‚€Β±βˆš(π‘₯)=2π‘₯𝑛𝑃𝑛+1±12√ξƒͺ,𝐽2π‘₯π‘›ξ‚€Β±βˆš(π‘₯)=2π‘₯𝑛𝐹𝑛+1±1√ξƒͺ,𝐽2π‘₯π‘›ξ‚€Β±βˆš(π‘₯)=ξ‚βˆ’2π‘₯𝑛𝐡𝑛±1√ξƒͺ,π½βˆ’2π‘₯βˆ’2π‘›ξ‚€Β±βˆš(π‘₯)=ξ‚βˆ’π‘₯𝑛Φ𝑛+1±1√ξƒͺ,π½βˆ’π‘₯𝑛(π‘₯)=22ξƒ©Β±ξ‚™βˆ’2π‘₯π‘Žξƒͺπ‘›π·π‘›ξ‚΅Β±ξ‚™π‘Žξ‚Ά.βˆ’2π‘₯,π‘Ž(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(π‘₯)=1 and π‘ž(π‘₯)=π‘₯ for constructing the Jacobsthal polynomials. Reference [20] also gave the inverse formula to present π‘ˆπ‘›(π‘₯) in terms of 𝐽𝑛(π‘₯). Actually, we can easily have the inverse formulas of π‘ˆπ‘›(π‘₯), 𝑃𝑛+1(π‘₯), 𝐹𝑛+1(π‘₯), Φ𝑛+1(π‘₯), and 𝐷𝑛(π‘₯,π‘Ž) in terms of 𝐽𝑛(π‘₯) as follows: π‘ˆπ‘›(π‘₯)=(2π‘₯)π‘›π½π‘›ξ‚€βˆ’18π‘₯2,𝑃𝑛+1(π‘₯)=(2π‘₯)𝑛𝐽𝑛18π‘₯2,𝐹𝑛+1(π‘₯)=π‘₯𝑛𝐽𝑛12π‘₯2,𝐡𝑛(π‘₯)=(π‘₯+2)π‘›π½π‘›ξ‚΅βˆ’12(π‘₯+2)2ξ‚Ά,Φ𝑛+1(π‘₯)=π‘₯π‘›π½π‘›ξ‚€βˆ’1π‘₯2,𝐷𝑛1(π‘₯,π‘Ž)=4π‘₯π‘›π½π‘›ξ‚€βˆ’π‘Ž2π‘₯2.(3.6)

Example 3.3. In Eu [22], the polynomial sequence {𝐻𝑛(π‘₯)} is defined by 𝑆𝑛(π‘₯)=π‘₯π‘†π‘›βˆ’1(π‘₯)βˆ’π‘†π‘›βˆ’2(π‘₯) with initial conditions 𝑆0(π‘₯)=1 and 𝑆1(π‘₯)=π‘₯. Using Corollary 2.5, we obtain 𝑆𝑛(π‘₯)=π‘ˆπ‘›ξ‚€Β±π‘₯2,𝑆𝑛(π‘₯)=(±𝑖)𝑛𝑃𝑛+1ξ‚€βˆ“π‘₯2𝑖,𝑆𝑛(π‘₯)=(±𝑖)𝑛𝐹𝑛+1𝑆(βˆ“π‘₯𝑖),𝑛(π‘₯)=(Β±1)𝑛𝐡𝑛𝑆(Β±π‘₯βˆ’2),𝑛(±1π‘₯)=√2ξƒͺ𝑛Φ𝑛+1ξ‚€Β±βˆšξ‚,𝑆2π‘₯𝑛±1(π‘₯)=4βˆšπ‘Žξƒͺπ‘›π·π‘›ξ‚€Β±βˆšξ‚,π‘Žπ‘₯,π‘Ž(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 π‘ˆπ‘›(π‘₯)=𝑆𝑛𝑃(Β±2π‘₯),𝑛+1(π‘₯)=(βˆ“π‘–)𝑛𝑆𝑛𝐹(Β±2π‘₯𝑖),𝑛+1(π‘₯)=(βˆ“π‘–)𝑛𝑆𝑛𝐡(Β±π‘₯𝑖),𝑛(π‘₯)=(Β±1)𝑛𝑆𝑛Φ(Β±(π‘₯+2)),𝑛+1ξ‚€Β±βˆš(π‘₯)=2𝑛𝑆𝑛±π‘₯√2ξƒͺ,𝐷𝑛1(π‘₯,π‘Ž)=4ξ‚€Β±βˆšπ‘Žξ‚π‘›π‘†π‘›ξƒ©Β±π‘₯βˆšπ‘Žξƒͺ.(3.8)

Another polynomial sequence {𝐻𝑛(π‘₯)} is defined by 𝐻𝑛(π‘₯)=(1βˆ’π‘₯)π»π‘›βˆ’1(π‘₯)βˆ’π‘₯2π»π‘›βˆ’2(π‘₯) with initial conditions 𝐻0(𝑋)=1 and 𝐻1(π‘₯)=1βˆ’π‘₯ [22]. Using Corollary 2.5, we obtain 𝐻𝑛(π‘₯)=(Β±π‘₯)π‘›π‘ˆπ‘›ξ‚€Β±1βˆ’π‘₯,𝐻2π‘₯𝑛(π‘₯)=(±𝑖π‘₯)𝑛𝑃𝑛+1ξ‚€βˆ“1βˆ’π‘₯𝑖,𝐻2π‘₯𝑛(π‘₯)=(±𝑖π‘₯)𝑛𝐹𝑛+1ξ‚€βˆ“1βˆ’π‘₯π‘₯𝑖,𝐻𝑛(π‘₯)=(Β±π‘₯)𝑛𝐡𝑛±1βˆ’π‘₯π‘₯,π»βˆ’2𝑛±π‘₯(π‘₯)=√2ξƒͺ𝑛Φ𝑛+1ξ‚€Β±βˆš21βˆ’π‘₯π‘₯,𝐻𝑛±π‘₯(π‘₯)=4βˆšπ‘Žξƒͺπ‘›π‘ˆπ‘›ξ‚€Β±βˆšπ‘Ž1βˆ’π‘₯π‘₯,,π‘Ž(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, π‘ˆπ‘›(π‘₯)=(2π‘₯Β±1)𝑛𝐻𝑛1ξ‚Ά1Β±2π‘₯.(3.10)

Example 3.4. In Riordan [23], the associate Legendre polynomial sequence {πœŒπ‘›(π‘₯)} is defined by πœŒπ‘›(π‘₯)=(2+π‘₯)πœŒπ‘›βˆ’1(π‘₯)βˆ’πœŒπ‘›βˆ’2(π‘₯) with initial conditions 𝜌0(π‘₯)=1 and 𝜌1(π‘₯)=1+π‘₯, then we use Theorem 2.1 and Corollary 2.2 to generate the following transfer formulas: πœŒπ‘›(π‘₯)=π‘ˆπ‘›ξ‚€Β±ξ‚€π‘₯1+2ξ‚ξ‚βˆ’π‘ˆπ‘›βˆ’1ξ‚€Β±ξ‚€π‘₯1+2,πœŒξ‚ξ‚π‘›(π‘₯)=(±𝑖)𝑛𝑃𝑛+1ξ‚€ξ‚€π‘₯βˆ“π‘–1+2ξ‚ξ‚βˆ’(±𝑖)π‘›βˆ’1𝑃𝑛π‘₯βˆ“π‘–1+2,πœŒξ‚ξ‚π‘›(π‘₯)=(±𝑖)𝑛𝐹𝑛+1(βˆ“π‘–(π‘₯+2))βˆ’(±𝑖)π‘›βˆ’1πΉπ‘›πœŒ(βˆ“π‘–(π‘₯+2)),𝑛(π‘₯)=(Β±1)𝑛𝐡𝑛(Β±(π‘₯+2)βˆ’2)βˆ’(Β±1)π‘›βˆ’1π΅π‘›βˆ’1𝜌(Β±(π‘₯+2)βˆ’2),𝑛(±1π‘₯)=√2ξƒͺ𝑛Φ𝑛+1ξ‚€Β±βˆšξ‚βˆ’ξƒ©Β±12(π‘₯+2)√2ξƒͺπ‘›βˆ’1Ξ¦π‘›ξ‚€Β±βˆšξ‚,𝜌2(π‘₯+2)𝑛±1(π‘₯)=4βˆšπ‘Žξƒͺπ‘›π·π‘›ξ‚€Β±βˆšξ‚ξƒ©Β±1π‘Ž(π‘₯+2),π‘Žβˆ’4βˆšπ‘Žξƒͺπ‘›βˆ’1π·π‘›βˆ’1ξ‚€Β±βˆšξ‚,π‘Ž(π‘₯+2),π‘Ž(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(π‘₯)} is defined by 𝑝𝑛(π‘₯)=βˆ’π‘₯π‘π‘›βˆ’1(π‘₯)βˆ’π‘₯π‘π‘›βˆ’2(π‘₯) with initial conditions 𝑝0(π‘₯)=1βˆ’π‘₯βˆ’1 and 𝑝1(π‘₯)=2βˆ’π‘₯ (π‘₯β‰ 0). From Theorem 2.1 and Corollary 2.2, we obtain 𝑝𝑛(π‘₯)=1βˆ’π‘₯βˆ’1ξ€Έξ‚€Β±βˆšπ‘₯ξ‚π‘›π‘ˆπ‘›ξƒ©βˆ“βˆšπ‘₯2ξƒͺ+ξ‚€Β±βˆšπ‘₯ξ‚π‘›βˆ’1π‘ˆπ‘›βˆ’1ξƒ©βˆ“βˆšπ‘₯2ξƒͺ,𝑝𝑛(π‘₯)=1βˆ’π‘₯βˆ’1ξ€Έξ‚€Β±βˆšξ‚π‘₯𝑖𝑛𝑃𝑛+1ξƒ©Β±βˆšπ‘₯𝑖2ξƒͺ+ξ‚€Β±βˆšξ‚π‘₯π‘–π‘›βˆ’1π‘ƒπ‘›ξƒ©Β±βˆšπ‘₯𝑖2ξƒͺ,𝑝𝑛(ξ€·π‘₯)=1βˆ’π‘₯βˆ’1ξ€Έξ‚€Β±βˆšξ‚π‘₯𝑖𝑛𝐹𝑛+1ξ‚€Β±βˆšξ‚+ξ‚€Β±βˆšπ‘₯𝑖π‘₯π‘–π‘›βˆ’1πΉπ‘›ξ‚€Β±βˆšξ‚,𝑝π‘₯𝑖𝑛(ξ€·π‘₯)=1βˆ’π‘₯βˆ’1ξ€Έξ‚€Β±βˆšπ‘₯ξ‚π‘›π΅π‘›ξ‚€βˆ“βˆšξ‚+ξ‚€Β±βˆšπ‘₯βˆ’2π‘₯ξ‚π‘›βˆ’1π΅π‘›βˆ’1ξ‚€βˆ“βˆšξ‚,𝑝π‘₯βˆ’2𝑛(π‘₯)=1βˆ’π‘₯βˆ’1ξ€Έξ‚΅Β±ξ‚™π‘₯2𝑛Φ𝑛+1ξ‚€Β±βˆšξ‚+ξ‚΅Β±ξ‚™2π‘₯𝑖π‘₯2ξ‚Άπ‘›βˆ’1Ξ¦π‘›ξ‚€Β±βˆšξ‚,𝑝2π‘₯𝑖𝑛(π‘₯)=41βˆ’π‘₯βˆ’1ξ€Έξ‚΅Β±ξ‚™π‘₯π‘Žξ‚Άπ‘›π·π‘›ξ‚€Β±βˆšξ‚ξ‚΅Β±ξ‚™π‘Žπ‘₯𝑖,π‘Ž+4π‘₯π‘Žξ‚Άπ‘›βˆ’1π·π‘›βˆ’1ξ‚€Β±βˆšξ‚.π‘Žπ‘₯𝑖,π‘Ž(3.12) Since π‘ˆπ‘›+1(𝑦)=2π‘¦π‘ˆπ‘›(𝑦)βˆ’π‘ˆπ‘›βˆ’1(𝑦), we have π‘ˆπ‘›+2(𝑦)=2π‘¦π‘ˆπ‘›+1(𝑦)βˆ’π‘ˆπ‘›ξ€·(𝑦)=2𝑦2π‘¦π‘ˆπ‘›(𝑦)βˆ’π‘ˆπ‘›βˆ’1ξ€Έ(𝑦)βˆ’π‘ˆπ‘›=ξ€·(𝑦)4𝑦2ξ€Έπ‘ˆβˆ’1𝑛(𝑦)βˆ’2π‘¦π‘ˆπ‘›βˆ’1(𝑦).(3.13) Hence, from the last expression of π‘ˆπ‘›+2 and the transfer formula of 𝑝𝑛(π‘₯) in terms of π‘ˆπ‘›(π‘₯) shown above, we obtain 𝑝𝑛(π‘₯)=(Β±1)𝑛π‘₯(π‘›βˆ’2)/2π‘ˆπ‘›+2ξƒ©βˆ“βˆšπ‘₯2ξƒͺ,(3.14) in which the case of 𝑝𝑛(π‘₯)=(βˆ’1)𝑛π‘₯(π‘›βˆ’2)/2π‘ˆπ‘›+2ξƒ©βˆšπ‘₯2ξƒͺ(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 π‘ˆπ‘›(π‘₯)βˆ’π‘₯π‘ˆπ‘›βˆ’1(π‘₯)=(βˆ’1)π‘›ξ€·π‘ˆπ‘›(βˆ’π‘₯)+π‘₯π‘ˆπ‘›βˆ’1ξ€Έ(π‘₯)=(±𝑖)𝑛𝑃𝑛+1(βˆ“π‘₯𝑖)βˆ’π‘₯(±𝑖)π‘›βˆ’1𝑃𝑛(βˆ“π‘₯𝑖)=(±𝑖)𝑛𝐹𝑛+1(βˆ“2π‘₯𝑖)βˆ’π‘₯(±𝑖)π‘›βˆ’1𝐹𝑛(βˆ“2π‘₯𝑖)=(Β±1)𝑛𝐡𝑛(Β±2π‘₯βˆ’2)βˆ’(Β±1)π‘›βˆ’1π‘₯π΅π‘›βˆ’1=±1(Β±2π‘₯βˆ’2)√2ξƒͺ𝑛Φ𝑛+1ξ‚€βˆšβˆ“2±12π‘₯βˆ’π‘₯√2ξƒͺπ‘›βˆ’1Ξ¦π‘›ξ‚€βˆšβˆ“2=±12π‘₯√ξƒͺ4π‘Žπ‘›π·π‘›ξ‚€βˆšβˆ“2±1π‘Žπ‘₯,π‘Žβˆ’π‘₯√ξƒͺ4π‘Žπ‘›βˆ’1π·π‘›βˆ’1ξ‚€βˆšβˆ“2.π‘Žπ‘₯,π‘Ž(3.16)

Using the relationship established in Theorem 2.1 and Corollaries 2.2–2.6, we may obtain some identities of polynomial sequences from the generalized Gegenbauer-Humbert polynomial sequence identity described in [5] 𝑃𝑛1,𝑦,𝐢(π‘₯)=𝛼(π‘₯)𝑃1,𝑦,πΆπ‘›βˆ’1(π‘₯)+πΆβˆ’2(2π‘₯βˆ’π›Ό(π‘₯)𝐢)(𝛽(π‘₯))π‘›βˆ’1,(3.17) where 𝑃𝑛1,𝑦,𝐢(π‘₯) satisfies the recurrence relation of order 2, 𝑃𝑛1,𝑦,𝐢(π‘₯)=𝑝(π‘₯)𝑃1,𝑦,πΆπ‘›βˆ’1(π‘₯)+π‘ž(π‘₯)𝑃1,𝑦,πΆπ‘›βˆ’2(π‘₯) with coefficients 𝑝(π‘₯) and π‘ž(π‘₯), and 𝛼(π‘₯)+𝛽(π‘₯)=𝑝(π‘₯) and 𝛼(π‘₯)𝛽(π‘₯)=βˆ’π‘ž(π‘₯). Clearly (see (2.41) in [5]), 1𝛼(π‘₯)=πΆξ‚†βˆšπ‘₯+π‘₯2,1βˆ’πΆπ‘¦π›½(π‘₯)=πΆξ‚†βˆšπ‘₯βˆ’π‘₯2.βˆ’πΆπ‘¦(3.18) For 𝑦=𝐢=1, we have 𝑃𝑛1,1,1(π‘₯)=π‘ˆπ‘›(π‘₯), where π‘ˆπ‘›(π‘₯) are the Chebyshev polynomials of the second kind, and we can write (3.17) as π‘ˆπ‘›(π‘₯)=𝛼(π‘₯)π‘ˆπ‘›βˆ’1(π‘₯)+(2π‘₯βˆ’π›Ό(π‘₯))(𝛽(π‘₯))π‘›βˆ’1=𝛼(π‘₯)π‘ˆπ‘›βˆ’1(π‘₯)+(𝛽(π‘₯))𝑛,(3.19) where βˆšπ›Ό(π‘₯)=π‘₯+π‘₯2βˆ’1 and βˆšπ›½(π‘₯)=π‘₯βˆ’π‘₯2βˆ’1. From the first formula of Example 3.2 and using transform ñ1/(2βˆ’2π‘₯)↦π‘₯, we have π‘ˆπ‘›(π‘₯)=(2π‘₯)π‘›π½π‘›ξ‚€βˆ’18π‘₯2.(3.20) Substituting the above expression to (3.19) yields the identity (2π‘₯)π‘›π½π‘›ξ‚€βˆ’18π‘₯2=ξ‚€βˆšπ‘₯+π‘₯2ξ‚βˆ’1(2π‘₯)π‘›βˆ’1π½π‘›βˆ’1ξ‚€βˆ’18π‘₯2+ξ‚€βˆšπ‘₯βˆ’π‘₯2ξ‚βˆ’1𝑛.(3.21) Similarly, from Example 3.3, we obtain identities 𝑆𝑛(ξ‚€βˆšΒ±2π‘₯)=Β±π‘₯+π‘₯2ξ‚π‘†βˆ’1π‘›βˆ’1(ξ‚€βˆšΒ±2π‘₯)+Β±π‘₯βˆ’π‘₯2ξ‚βˆ’1𝑛,(2π‘₯Β±1)𝑛𝐻𝑛1ξ‚Ά1Β±2π‘₯=(2π‘₯Β±1)π‘›βˆ’1ξ‚€βˆšπ‘₯+π‘₯2ξ‚π»βˆ’1π‘›βˆ’1ξ‚΅1ξ‚Ά+ξ‚€βˆš1Β±2π‘₯π‘₯βˆ’π‘₯2ξ‚βˆ’1𝑛.(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(π‘₯)πΉπ‘›βˆ’1(π‘₯)βˆ’πΉ2𝑛(π‘₯)=(βˆ’1)𝑛,(3.23) we use the relationship shown in Example 3.2 to obtain the Cassini-like formula for the Jacobsthal polynomials 𝐽𝑛(π‘₯)π½π‘›βˆ’2(π‘₯)βˆ’π½2π‘›βˆ’1(π‘₯)=(βˆ’2π‘₯)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(π‘₯)=(±𝑖)π‘›π‘ˆπ‘›ξ‚€βˆ“π‘₯𝑖2,(3.25) we have π‘ˆπ‘›ξ‚€βˆ“π‘₯𝑖2ξ‚π‘ˆπ‘›βˆ’2ξ‚€βˆ“π‘₯𝑖2ξ‚βˆ’π‘ˆ2π‘›βˆ’1ξ‚€βˆ“π‘₯𝑖2=(βˆ’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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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.

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