# Sequences of Numbers Meet the Generalized Gegenbauer-Humbert Polynomials

Authors: Tian-Xiao He; Peter J.-S. Shiue; Tsui-Wei Weng
Journal: ISRN Discrete Mathematics
 (2011)

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## Abstract

Here we present a connection between a sequence of numbers generated by a linear recurrence relation of order 2 and sequences of the generalized Gegenbauer-Humbert polynomials. Many new and known formulas of the Fibonacci, the Lucas, the Pell, and the Jacobsthal numbers in terms of the generalized Gegenbauer-Humbert polynomial values are given. The applications of the relationship to the construction of identities of number and polynomial value sequences defined by linear recurrence relations are also discussed.


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## Body

## 1. Introduction

Many number and polynomial sequences can be defined, characterized, evaluated, and/or classified by linear recurrence relations with certain orders. A number sequence{an} is called sequence of order 2 if it satisfies the linear recurrence relation of order 2:
(1.1)an=pan-1+qan-2,n≥2,
for some nonzero constants p and q and initial conditions a0 and a1. In Mansour [1], the sequence {an}n≥0 defined by (1.1) is called Horadam’s sequence, which was introduced in 1965 by Horadam [2]. In [1] also the generating functions for powers of Horadam’s sequence are obtained. To construct an explicit formula of its general term, one may use a generating function, characteristic equation, or a matrix method (see Comtet [3], Hsu [4], Strang [5], Wilf [6], etc.) In [7], Benjamin and Quinn presented many elegant combinatorial meanings of the sequence defined by recurrence relation (1.1). For instance, an counts the number of ways to tile an n-board (i.e., board of length n) with squares (representing 1s) and dominoes (representing 2s) where each tile, except the initial one, has a color. In addition, there are p colors for squares and q colors for dominoes. In particular, Aharonov et al. (see [8]) 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 the Chebyshev polynomial values. For instance, the authors show Fn=i-nUn(i/2) and Ln=2i-nTn(i/2), where Fn and Ln, respectively, are the Fibonacci numbers and Lucas numbers, and Tn and Un are the Chebyshev polynomials of the first kind and the second kind, respectively. Some identities drawn from those relations were given by Beardon in [9]. Marr and Vineyard in [10] use the relationship to establish explicit expression of five-diagonal Toeplitz determinants. In [11], the first two authors presented a new method to construct an explicit formula of {an} generated by (1.1). For the sake of the reader's convenience, we cite this result as follows.Proposition 1.1 (see [11]).
Let{an} be a sequence of order 2 satisfying linear recurrence relation (1.1), and let α and β be two roots of of quadratic equation x2-px-q=0. Then
(1.2)an={(a1-βa0α-β)αn-(a1-αa0α-β)βn,ifα≠β,na1αn-1-(n-1)a0αn,ifα=β.A sequence of the generalized Gegenbauer-Humbert polynomials{Pnλ,y,C(x)}n≥0 is defined by the expansion (see, e.g., Comtet [3], Gould [12], Lidl et al. [13], the two authors with He et al. [14])
(1.3)Φ(t)≡(C-2xt+yt2)-λ=∑n≥0Pnλ,y,C(x)tn,
where λ>0, y and C≠0 are real numbers. As special cases of (1.3), we consider Pnλ,y,C(x) as follows (see [14]): Pn1,1,1(x)=Un(x), the Chebyshev polynomial of the second kind,Pn1/2,1,1(x)=ψn(x), the Legendre polynomial,Pn1,-1,1(x)=Pn+1(x), the Pell polynomial,Pn1,-1,1(x/2)=Fn+1(x), the Fibonacci polynomial,Pn1,2,1(x/2)=Φn+1(x), the Fermat polynomial of the first kind,Pn1,2a,2(x)=Dn(x,a), the Dickson polynomial of the second kind, a≠0 (see, e.g., [13]),wherea is a real parameter, and Fn=Fn(1) is the Fibonacci number. In particular, if y=C=1, the corresponding polynomials are called the Gegenbauer polynomials (see [3]). More results on the Gegenbauer-Humbert-type polynomials can be found in [15] by Hsu and in [16] by the second author and Hsu, and so forth.Similarly, for a class of the generalized Gegenbauer-Humbert polynomial sequences defined by(1.4)Pnλ,y,C(x)=2xλ+n-1CnPn-1λ,y,C(x)-y2λ+n-2CnPn-2λ,y,C(x),
for all n≥2 with initial conditions(1.5)P0λ,y,C(x)=Φ(0)=C-λ,P1λ,y,C(x)=Φ′(0)=2λxC-λ-1,
the following theorem has been obtained in [11].Theorem 1.2 (see [11]).
Letx≠±Cy. The generalized Gegenbauer-Humbert polynomials {Pn1,y,C(x)}n≥0 defined by expansion (1.3) can be expressed as
(1.6)Pn1,y,C(x)=C-n-2(x+x2-Cy)n+1-(x-x2-Cy)n+12x2-Cy.In this paper, we will use an alternative form of (1.2) to establish a relationship between the number sequences defined by recurrence relation (1.1) and the generalized Gegenbauer-Humbert polynomial sequences defined by (1.4). Our results are suitable for all such number sequences defined by (1.1) with arbitrary initial conditions a0 and a1, which includes the results in [8, 9] as our special cases. Many new and known formulas of the Fibonacci, the Lucas, the Pell, and the Jacobsthal numbers in terms of the generalized Gegenbauer-Humbert polynomial values and applications of the established relationship to the construction of identities of number and polynomial value sequences will be presented in Section 3.


## 2. Main Results

We now modify the explicit formula of the number sequences defined by linear recurrence relations of order 2. Ifα≠β, the first formula in (1.2) can be written as
(2.1)an=a1(αn-βn)-a0αβ(αn-1-βn-1)α-β=a1(αn-βn)+a0q(αn-1-βn-1)α-β,
where the last step is due to α and β being solutions of t2-pt-q=0. Noting that α2-pα=α2-(α+β)α=-αβ=q and α(α-p)=-αβ=β(β-p), we may further write the above last expression of an as
(2.2)an=a1(αn-βn)+a0(α2-pα)(αn-1-βn-1)α-β=a1(αn-βn)+a0(α2-pα)αn-1-a0(β2-pβ)βn-1α-β=a0(αn+1-βn+1)+(a1-a0p)(αn-βn)α-β.Denoter(x)=x+x2-Cy and s(x)=x-x2-Cy. Comparing expressions (2.2) and (1.6), we have reason to consider the following transform: for a nonzero real or complex number k, we set
(2.3)α:=r(x)k,β:=s(x)k
for a certain x depending on α, β, and k, which we will find out later. Denote α+β=p and αβ=-q; that is, α and β are roots of t2-pt-q. By adding the two equations in (2.3) side by side, we obtain 2x=kp. Thus, when x=kp/2, the equations in (2.2) hold. Meanwhile, by using (α-β)2=(α+β)2-4αβ=p2+4q, we have
(2.4)r(x)-s(x)=2x2-Cy=k(α-β)=kp2+4q,
where x=kp/2. Therefore, we obtain
(2.5)2(kp2)2-Cy=kp2+4q,
which implies
(2.6)k=±Cy-q.
We first consider the case of k=-Cy/q.We now substituter(x)=kα, s(x)=kβ, x=kp/2, and k=-Cy/q into (2.2) and simplify as follows:(2.7)an=a0((r(x)/k)n+1-(s(x)/k)n+1)+(a1-a0p)((r(x)/k)n-(s(x)/k)n)(1/k)(r(x)-s(x))=a0(rn+1(x)-sn+1(x))+k(a1-a0p)(rn(x)-sn(x))kn(r(x)-s(x))=a0Cn+2(-qCy)nPn1,y,C(kp2)+(a1-a0p)Cn+1(-qCy)n-1Pn-11,y,C(kp2)=a0Cn+2(-qCy)nPn1,y,C(p2Cy-q)+(a1-a0p)Cn+1(-qCy)n-1Pn-11,y,C(p2Cy-q).
Similarly, for k=--Cy/q, we have
(2.8)an=a0Cn+2(--qCy)nPn1,y,C(-p2Cy-q)+(a1-a0p)Cn+1(--qCy)n-1Pn-11,y,C(-p2Cy-q).Therefore, we obtain our main result.Theorem 2.1.
Let sequence{an} be defined by an=pan-1+qan-2 (n≥2) with initial conditions a0 and a1. Then, an can be presented as (2.7) and (2.8). In particular, for (y,C)=(1,1),(-1,1),(2,1), and (2a,2)(a≠0), respectively, one has
(2.9)an=a0(-q)nUn(p2-q)+(a1-a0p)(-q)n-1Un-1(p2-q),an=a0(q)nPn+1(p2q)+(a1-a0p)(q)n-1Pn(p2q),an=a0(q)nFn+1(pq)+(a1-a0p)(q)n-1Fn(pq),an=a0(-q2)nΦn+1(p2-q)+(a1-a0p)(-q2)n-1Φn(p2-q),an=a02n+2(-q4a)nDn(pa-q,a)+(a1-a0p)2n+1(-q4a)n-1Dn-1(pa-q,a),an=a0(--q)nUn(-p2-q)+(a1-a0p)(--q)n-1Un-1(-p2-q),an=a0(-q)nPn+1(-p2q)+(a1-a0p)(-q)n-1Pn(-p2q),an=a0(-q)nFn+1(-pq)+(a1-a0p)(-q)n-1Fn(-pq),an=a0(--q2)nΦn+1(-p2-q)+(a1-a0p)(--q2)n-1Φn(-p2-q),an=a02n+2(--q4a)nDn(-pa-q,a)+(a1-a0p)2n+1(--q4a)n-1Dn-1(-pa-q,a),
where Un(x), Pn(x), Fn(x), Φn(x), and Dn(x,a) are the nth degree Chebyshev polynomial of the second kind, the Pell polynomial, the Fibonacci polynomial, the Fermat polynomial, and the Dickson polynomial of the second kind, respectively.For the special cases ofa0 and a1, we have the following corollaries.Corollary 2.2.
Let sequence{an} be defined by an=pan-1+qan-2 (n≥2) with initial conditions a0=0 and a1=d. Then
(2.10)an=d(-q)n-1Un-1(p2-q),an=d(q)n-1Pn(p2q),an=d(q)n-1Fn(pq),an=d(-q2)n-1Φn(p2-q),an=d2n+1(-q4a)n-1Dn-1(pa-q,a),an=d(--q)n-1Un-1(-p2-q),an=d(-q)n-1Pn(-p2q),an=d(-q)n-1Fn(-pq),an=d(--q2)n-1Φn(-p2-q),an=d2n+1(--q4a)n-1Dn-1(-pa-q,a).Corollary 2.3.
Let sequence{an} be defined by an=pan-1+qan-2 (n≥2) with initial conditions a0=c and a1=pc. Then
(2.11)an=c(-q)nUn(p2-q),an=c(q)nPn+1(p2q),an=c(q)nFn+1(pq),an=c(-q2)nΦn+1(p2-q),an=c2n+2(-q4a)nDn(pa-q,a),an=c(--q)nUn(-p2-q),an=c(-q)nPn+1(-p2q),an=c(-q)nFn+1(-pq),an=c(--q2)nΦn+1(-p2-q),an=c2n+2(--q4a)nDn(-pa-q,a).Ifa1=d=1, then Corollary 2.2 gives the primary solutions of recurrence relation (1.1) in terms of the nth degree Chebyshev polynomial of the second kind, the Pell polynomial, the Fibonacci polynomial, the Fermat polynomial, and the Dickson polynomial of the second kind, respectively. For instance, if p=q=1, then an are the Fibonacci numbers Fn. Thus,
(2.12)Fn=(i)n-1Un-1(12i)=(i)n-1Un-1(-i2),Fn=Pn(12),Fn=Fn(1),Fn=(i2)n-1Φn(-2i),Fn=2n+1(i4a)n-1Dn-1(-ai,a),Fn=(-i)n-1Un-1(i2),Fn=(-1)n-1Pn(-12),Fn=(-1)n-1Fn(-1),Fn=(-i2)n-1Φn(2i),Fn=2n+1(-i4a)n-1Dn-1(ai,a),
where Fn=(i)n-1Un-1(-i/2) was shown in [8] and Fn=(-i)n-1Un-1(i/2) was given by Chen and Louck in [17]. From the above expressions of Fn, we may obtain many identities. For instance, we have
(2.13)Pn(12)=(-1)n-1Pn(-12)=Fn(1)=(-1)n-1Fn(-1),(i)n-1Un-1(-i2)=(-i)n-1Un-1(i2)=(i2)n-1Φn(-2i)=(-i2)n-1Φn(2i),
and so forth.We now give another special case of Theorem2.1 for the sequence defined by (1.1) with initial cases a0=2 and a1.Corollary 2.4.
Let sequence{an} be defined by an=pan-1+qan-2 (n≥2) with initial conditions a0=2 and a1=p. Then
(2.14)an=2(-q)nUn(p2-q)-p(-q)n-1Un-1(p2-q),an=2(q)nPn+1(p2q)-p(q)n-1Pn(p2q),an=2(q)nFn+1(pq)-p(q)n-1Fn(pq),an=2(-q2)nΦn+1(p2-q)-p(-q2)n-1Φn(p2-q),an=2n+3(-q4a)nDn(pa-q,a)-p2n+1(-q4a)n-1Dn-1(pa-q,a),an=2(--q)nUn(-p2-q)-p(--q)n-1Un-1(-p2-q),an=2(-q)nPn+1(-p2q)-p(-q)n-1Pn(-p2q),an=2(-q)nFn+1(-pq)-p(-q)n-1Fn(-pq),an=2(--q2)nΦn+1(-p2-q)-p(--q2)n-1Φn(-p2-q),an=2n+3(--q4a)nDn(-pa-q,a)-p2n+1(--q4a)n-1Dn-1(-pa-q,a).
In addition, one has
(2.15)an=2(-q)nTn(p2-q),(2.16)an=2(--q)nTn(-p2-q),
where Tn(x) are the Chebyshev polynomials of the first kind.Proof.
It is sufficient to prove (2.15) and (2.16). From the first formula shown in Corollary 2.4 and the recurrence relation Un(x)=2xUn-1(x)-Un-2(x), one easily sees
(2.17)an=(-q)n[2Un(p2-q)-p-qUn-1(p2-q)]=(-q)n[2Un(p2-q)-(Un(p2-q)+Un-2(p2-q))]=(-q)n[Un(p2-q)-Un-2(p2-q)].
From the basic relation between Chebyshev polynomials of the first and the second kinds (see, e.g., (1.7) in [18] by Mason and Handscomb), Un(x)-Un-2(x)=2Tn(x), the last expression of an implies (2.15). Equation (2.16) can be proved similarly.As an example, the Lucas number sequence{Ln} defined by (1.1) with p=q=1 and initial conditions L0=2 and L1=1 has the explicit formula for its general term:
(2.18)Ln=2inTn(-i2)=2(-i)nTn(i2).


## 3. Examples and Applications

We first give some examples of Corollary2.2 for sequences {an} that are primary solutions of (1.1).Example 3.1.
Ifp=2 and q=1, then an defined by (1.1) with initial conditions a0=0  and  a1=1 are the Pell numbers Pn. Thus, from Corollary 2.2, we have
(3.1)Pn=(i)n-1Un-1(-i)=(-i)n-1Un-1(i),Pn=Pn(1)=(-1)n-1Pn(-1),Pn=Fn(2)=(-1)n-1Fn(-2),Pn=(i2)n-1Φn(-22i)=(-i2)n-1Φn(22i),Pn=2n+1(i4a)n-1Dn-1(-2ai,a)=2n+1(-i4a)n-1Dn-1(2ai,a).Example 3.2.
Ifp=1 and q=2, then an defined by (1.1) with initial conditions a0=0 and a1=1 are the Jacobsthal numbers Jn (see Bergum et al. [19]). Thus Corollary 2.2 gives the expressions of Jn as follows:
(3.2)Jn=(2i)n-1Un-1(-i22)=(-2i)n-1Un-1(i22),Jn=(2)n-1Pn(122)=(-2)n-1Pn(-122),Jn=(2)n-1Fn(12)=(-2)n-1Fn(-12),Jn=in-1Φn(-pi)=(-i)n-1Φn(pi),Jn=2n+1(i2a)n-1Dn-1(-pai2,a)=2n+1(-i2a)n-1Dn-1(pai2,a).Example 3.3.
Ifp=3 and q=-2, then an defined by (1.1) with initial conditions a0=0 and a1=1 are the Mersenne numbers Mn=2n-1. From Corollary 2.2, we have
(3.3)Mn=(2)n-1Un-1(322)=(-2)n-1Un-1(-322),Mn=(2i)n-1Pn(-3i22)=(-2i)n-1Pn(3i22),Mn=(2i)n-1Fn(-3i2)=(-2i)n-1Fn(3i2),Mn=Φn(3)=(-1)n-1Φn(-3),Mn=2n+1(12a)n-1Dn-1(3a2,a)=2n+1(-12a)n-1Dn-1(-3a2,a).Next, we give several examples of nonprimary solutions of (1.1) by using Corollary 2.4.Example 3.4.
Ifp=1 and q=1, then an defined by (1.1) with initial conditions a0=2 and a1=1 are the Lucas numbers Ln. Thus, besides (2.18), we have
(3.4)Ln=2inUn(-i2)-in-1Un-1(-i2)=2(-i)nUn(i2)-(-i)n-1Un-1(i2),Ln=2Pn+1(12)-Pn(12)=2(-1)nPn+1(-12)-(-1)n-1Pn(-12),Ln=2Fn+1(1)-Fn(1)=2(-1)nFn+1(-1)-(-1)n-1Fn(-1),Ln=2(i2)nΦn+1(-2i)-(i2)n-1Φn(-2i)=2(-i2)nΦn+1(2i)-(-i2)n-1Φn(2i),Ln=2n+3(i4a)nDn(-ai,a)-2n+1(i4a)n-1Dn-1(-ai,a)=2n+3(-i4a)nDn(ai,a)-2n+1(-i4a)n-1Dn-1(ai,a).Example 3.5.
Ifp=2 and q=1, then an defined by (1.1) with initial conditions a0=2 and a1=2 are the Pell-Lucas numbers An (see Example 2 in [11]). Thus, from Corollary 2.4, we obtain
(3.5)An=2inTn(-i)=2(-i)nTn(i),An=2inUn(-i)-2in-1Un-1(-i)=2inUn(-i)-2in-1Un-1(-i),An=2Pn+1(1)-2Pn(1)=2(-1)nPn+1(-1)-p(-1)n-1Pn(-1),An=2Fn+1(2)-2Fn(2)=2(-1)nFn+1(-2)-p(-1)n-1Fn(-2),An=2(-i2)nΦn+1(22i)-2(-i2)n-1Φn(22i)=2(i2)nΦn+1(-22i)-2(i2)n-1Φn(-22i),An=2n+3(i4a)nDn(-2ai,a)-2n+2(i4a)n-1Dn-1(-2ai,a)=2n+3(-i4a)nDn(2ai,a)-2n+2(-i4a)n-1Dn-1(2ai,a).Example 3.6.
Ifp=1 and q=2, then an defined by (1.1) with initial conditions a0=2 and a1=1 are the Jacobsthal-Lucas numbers Bn (see Example 2 in [11]). Thus,
(3.6)Bn=2(2i)nTn(-i22)=2(-2i)nTn(i22),Bn=2(2i)nUn(-i22)-(2i)n-1Un-1(-i22)=2(-2i)nUn(i22)-(-2i)n-1Un-1(i22),Bn=2(2)nPn+1(122)-(2)n-1Pn(122)=2(-2)nPn+1(-122)-(-2)n-1Pn(-122),Bn=2(2)nFn+1(12)-(2)n-1Fn(12)=2(-2)nFn+1(-12)-(-2)n-1Fn(-12),Bn=2inΦn+1(-i)-in-1Φn(-i)=2(-i)nΦn+1(i)-(-i)n-1Φn(i),Bn=2n+3(i2a)nDn(-ai2,a)-2n+1(i2a)n-1Dn-1(-ai2,a)=2n+3(-i2a)nDn(ai2,a)-2n+1(-i2a)n-1Dn-1(ai2,a).Example 3.7.
Ifp=3 and q=-2, then an defined by (1.1) with initial conditions a0=2 and a1=3 are the Fermat numbers fn (see [20]). Thus, from Corollary 2.4, we obtain
(3.7)fn=2(2)nTn(322)=2(-2)nTn(-322),fn=2(2)nUn(322)-3(2)n-1Un-1(322)=2(-2)nUn(-322)-3(-2)n-1Un-1(-322),fn=2(2i)nPn+1(-3i22)-3(2i)n-1Pn(-3i22)=2(-2i)nPn+1(3i22)-3(-2i)n-1Pn(3i22),fn=2(2i)nFn+1(-3i2)-3(2i)n-1Fn(-3i2)=2(-2i)nFn+1(3i2)-3(-2i)n-1Fn(3i2),fn=2Φn+1(3)-3Φn(3)=2(-1)nΦn+1(-3)-3(-1)n-1Φn(-3),fn=2n+3(12a)nDn(3a2,a)-(3)2n+1(12a)n-1Dn-1(3a2,a)=2n+3(-12a)nDn(-3a2,a)-(3)2n+1(-12a)n-1Dn-1(-3a2,a).Using the relationship established above, we may obtain some identities of number sequences and polynomial value sequences. Theorem 3.2 in [11] presented a generalized Gegenbauer-Humbert polynomial sequence identity:
(3.8)Pn1,y,C(x)=α(x)Pn-11,y,C(x)+C-2(2x-α(x)C)(β(x))n-1,
where Pn1,y,C(x) satisfies the recurrence relation of order 2, Pn1,y,C=pPn-11,y,C+qPn-21,y,C with coefficients p(x) and q(x), and α(x)+β(x)=p(x) and α(x)β(x)=-q(x). Clearly (see (19) and (20) in [11]),
(3.9)α=1C{x+x2-Cy},β=1C{x-x2-Cy}.
For y=-1 and C=1, we have Pn1,-1,1(x)=Fn+1(2x), where Fn(x) are the Fibonacci polynomials, and we can write (3.8) as
(3.10)Fn+1(2x)=α(x)Fn(2x)+(2x-α(x))(β(x))n-1=α(x)Fn(2x)+(β(x))n,
where α(x)=x+x2+1 and β(x)=x-x2+1. If x=1/2, then Fn(1)=Fn, the Fibonacci numbers, and
(3.11)α(12)=1+52,β(12)=1-52.Thus (3.10) yields the identity
(3.12)Fn+1=1+52Fn+(1-52)n,
or equivalently,
(3.13)1-52Fn+1+Fn=(1-52)n+1.Similarly, ifx=1, then Fn(2)=Pn, the Pell numbers, and
(3.14)α(1)=1+2,β(1)=1-2.
Thus (3.10) yields the identity(3.15)Pn+1=(1+2)Pn+(1-2)n,
or equivalently,
(3.16)(1-2)Pn+1+Pn=(1-2)n+1.Substitutingx=1/(22) into (3.10) and noting Fn(1/2)=Jn/(2)n, where Jn are the Jacobsthal numbers, we obtain the identity
(3.17)Jn+1-2Jn=(-1)n.
When x=-3i/(22), Fn(-3i/(22))=Mn/(2i)n-1, the Mersenne numbers. Hence (3.10) gives Mn+1-Mn=2n.Conversely, one may use the expressions of various number sequences in terms of the generalized Gegenbauer-Humbert polynomial sequences to construct the identities of the different generalized Gegenbauer-Humbert polynomial values such as the formulas shown in the example after Corollary2.3.



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