Linearization formulae for certain Jacobi polynomials
W. M. Abd-Elhameed, E. H. Doha & H. M. Ahmed
The Ramanujan Journal An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan ISSN 1382-4090 Ramanujan J DOI 10.1007/s11139-014-9668-2
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Author's personal copy Ramanujan J DOI 10.1007/s11139-014-9668-2
Linearization formulae for certain Jacobi polynomials W. M. Abd-Elhameed · E. H. Doha · H. M. Ahmed
Received: 21 May 2014 / Accepted: 26 December 2014 © Springer Science+Business Media New York 2015
Abstract In this article, some new linearization formulae of products of Jacobi polynomials for certain parameters are derived. These new derived formulae are expressed in terms of hypergeometric functions of unit argument, and they generalize some existing formulae in the literature. With the aid of some standard formulae and also by employing symbolic algebraic computation, and in particular Zeilberger’s algorithm, several reduction formulae for summing certain terminating hypergeometric functions of unit argument are given, and hence several linearization formulae of products of Jacobi polynomials for special parameters free of hypergeometric functions are deduced. Keywords Jacobi polynomials · Linearization problems · Generalized hypergeometric functions · Algorithms by Zeilberger, Petkovsek and van Hoeij Mathematics Subject Classification
42C10 · 33A50 · 33C25 · 33D45
W. M. Abd-Elhameed (B) Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia e-mail:
[email protected] W. M. Abd-Elhameed · E. H. Doha Department of Mathematics, Faculty of Science, Cairo University, Cairo, Egypt E. H. Doha e-mail:
[email protected] H. M. Ahmed Department of Mathematics, Faculty of Industrial Education, Helwan University, Cairo, Egypt e-mail:
[email protected] H. M. Ahmed Department of Mathematics, Faculty of Science, Shaqra University, Shaqra, Saudi Arabia
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1 Introduction The Jacobi polynomials play a prominent role in theoretical mathematical analysis as well as in applied mathematical analysis. There are six well-known families in the general class of Jacobi polynomials. The Jacobi polynomials in general and their special cases in particular are very useful for applications in various disciplines. For example, they have been extensively utilized in spectral methods for solving ordinary and partial differential equations of integer and fractional orders (see, for instance, [1,16]). From the theoretical point of view, the two problems of linearizing products of orthogonal polynomials and finding the connection coefficients between them are of fundamental interest. In particular, the linearization and connection coefficient problems for ultraspherical and Jacobi polynomials have been studied by a large number of authors: Chaggara and Koepf [4], Doha [6–8], Doha and Abd-Elhameed [9], Doha and Ahmed [10], Maroni and da Rocha [20], Sánchez-Ruiz [26], Sánchez-Ruiz and Dehesa [27] and Sánchez-Ruiz et al. [28]. Recently, Tcheutia [29] has discussed in his Ph.D. thesis, the problems of connection, linearization and duplication coefficients of classical orthogonal polynomials. In general, if we have two polynomials Pi (x) and Q j (x) of degrees i and j, respectively, then the linearization problem consists of finding the coefficients Ai, j,k in the expansion of the product of the two polynomials Pi (x) and Q j (x) in terms of an arbitrary sequence of orthogonal polynomials {Rk (x)}k≥0 , i.e., Pi (x) Q j (x) =
i+ j
Ai, j,k Rk (x).
(1)
k=0
The following two problems are important particular cases of the general linearization problem (1): (i) The standard linearization or Clebsch–Gordan-type problem which consists of finding the coefficients L i, j,k in the expansion of the product of two polynomials Pi (x) and P j (x) in terms of the sequence polynomials {Pn }n≥0 , i.e., Pi (x) P j (x) =
i+ j
L i, j,k Pk (x).
(2)
k=0
(ii) The connection problem which consists of finding the coefficients Bi,k such that Pi (x) =
i
Bi,k Rk (x).
k=0
Linearization problems arise in many applications. The special case in which i = j in the standard linearization formula (2) is useful when evaluating the logarithmic potentials of orthogonal polynomials appearing in the calculation of the position and momentum information entropies of quantum systems (see, Dehesa et al. [5]).
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The standard linearization problem associated with Jacobi polynomials to establish the conditions of nonnegativity of the linearization coefficients has been studied by many authors (see, for instance, Askey and Gasper [3], Gasper [13,14], Hylleraas [15], and Rahman [24]). The main objective of this article is to derive some new linearization formulae of products of Jacobi polynomials for certain parameters. In particular, we find the (α,β) (β,α) (x), Q j (x) = P j (x), and Rk (x) = coefficients Ai, j,k in (1) when Pi (x) = Pi (ν)
Ck (x) (a Gegenbauer polynomial). The contents of the paper are as follows. In Sect. 2, some preliminaries are given including some relevant properties of Jacobi polynomials and also some transformation formulae of hypergeometric functions. In Sect. 3, we state and prove our main theorem, in which a new linearization formula of products of some Jacobi polynomials for certain parameters is given. Some important special formulae, obtained from the formula given in Sect. 3 by making use of some standard formulae and also by employing some computer algebra algorithms such as the algorithms of Zeilberger, Petkovsek and van Hoeij, are developed and deduced in Sect. 4. 2 Preliminaries This section is devoted to introducing some properties of Jacobi polynomials which will be useful throughout the paper. Moreover, some important transformation formulae between some types of hypergeometric functions are presented. 2.1 Some properties of the classical Jacobi polynomials It is well known that the classical Jacobi polynomials associated with the real parameters (α > −1 , β > −1) and the weight function w(x) = (1 − x)α (1 + x)β , (see, for instance, Andrews et al. [2] and Olver et al. [23]), are a sequence of polynomials (α,β) (x) (n = 0, 1, 2, . . .), x ∈ [−1, 1], each, respectively, of degree n. The explicit Pn Gauss hypergeometric representation for these polynomials is Pn(α,β) (x)
(α + 1)n = 2 F1 n! (−1)n (β + 1)n = n!
−n, n + λ 1 − x 2 α+1 −n, n + λ 1 + x , 2 F1 2 β +1
(3)
where λ = α + β + 1, (a)k =
(a + k) . (a) (α,β)
For the sake of simplicity, we introduce the orthogonal Jacobi polynomials Rn x ∈ [−1, 1], (see, [11])
(x),
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Rn(α,β) (x)
(α,β)
(x)
n! = = (α,β) P (α,β) (x) = 2 F1 (α + 1)n n Pn (1) Pn
−n, n + λ 1 − x . (4) 2 α+1
These polynomials are orthogonal on [−1, 1] with respect to the weight function (1 − x)α (1 + x)β , in the sense that
1
−1
(α,β) (1 − x)α (1 + x)β Rm (x) Rn(α,β) (x) d x
⎧ ⎨0, = 2λ n! (n + β + 1) [(α + 1)]2 ⎩ , (2n + λ) (n + λ) (n + α + 1)
m = n, (5)
m = n.
The definition in (4) enables one to obtain the following six special polynomials of (α,β) Rn (x) in the following explicit forms: (α− 21 ,α− 12 )
Cn(α) (x) = Rn
(− 21 ,− 21 )
(x),
( 12 , 12 )
Un (x) = (n + 1) Rn
Tn (x) = Rn
(− 21 , 21 )
(x),
( 12 ,− 12 )
Wn (x) = (2n + 1) Rn
Vn (x) = Rn
(x),
(x),
Pn (x) = Rn(0,0) (x),
(x),
(α)
where Cn (x), Tn (x), Un (x), Vn (x), Wn (x) and Pn (x) are the ultraspherical, Chebyshev of the first, second, third and fourth kinds and Legendre polynomials, respectively. For properties of the four kinds of Chebyshev polynomials, one can be referred to the important book of Mason and Handscomb [21].
2.2 Some transformation formulae In this subsection, we present some transformation formulae which will be very useful in the sequel. First, we recall the definition of the generalized hypergeometric function
p Fq
∞ (a p )k z k
a p z = , bq (bq )k k! k=0
where the symbols a p and bq denote, respectively, to the two sets a1 , a2 , . . . , a p and b1 , b2 , . . . , bq of real or complex parameters, where b j = 0, for all 1 ≤ j ≤ q, p q and (a p )k = (ai )k , (bq )k = (b j )k . i=1
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Lemma 1 For all nonnegative integers n, p, q, r, s, t, u, one has
n a p k (αt )k z k n −n, a p , [cr ]
z w = p+r +1 Fq+s ¯ k bq , [ds ] k bq k (βu )k (k + λ) k=0
k − n, k + a p , [k + αt ] z × p+t+1 Fq+u+1 2 k + λ¯ + 1, k + bq , [k + βu ] −k, k + λ¯ , [cr ] , [βu ] × r +u+2 Fs+t w . [ds ] , [αt ]
(6)
(For the proof of Lemma 1, see Fields and Wimp [12], and Luke [19]). Also, the following transformation formula is essential (see, Rainville [25]): a, b a, b z z F 2 1 c 1+a+b−c a, b, 21 (a + b), 21 (a + b + 1) 4z(1 − z) . = 4 F3 a + b, c, 1 + a + b − c
2 F1
(7)
The following theorem is also useful in the sequel. Theorem 1 Pfaff–Saalschütz identity (see, Olver et al. [23]) For every nonnegative integer n, and for c + d = a + b + 1 − n, one has 3 F2
(c − a)n (c − b)n −n, a, b 1 = . c, d (c)n (c − a − b)n
(8)
3 New linearization formulae of some Jacobi polynomials In this section, a new linearization formula of product of Jacobi polynomials for certain parameters is given in terms of ultraspherical polynomials. In this respect, we state and prove the following theorem. Theorem 2 For every nonnegative integer n and for all α > −1, β > −1 and ν > − 21 , the following linearization formula holds Rn(α,β) (x) ×5 F4
Rn(β,α) (x)
=
n
n ν + 21 k λ2 k λ+1 k 2 k (n + λ)k
(α + 1)k (β + 1)k (k + ν)k (λ)k 1 k − n, k + n + λ, k + λ2 , k + λ+1 2 , k + ν + 2 1 C (ν) (x). 2k k + α + 1, k + β + 1, k + λ, 2k + ν + 1 k=0
(9)
Proof If we make use of the hypergeometric form of the introduced Jacobi polynomial (α,β) Rn (x) in (4), then we get
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Rn(α,β) (x) Rn(β,α) (x) = 2 F1
−n, n + λ 1 − x α+1 2
2 F1
−n, n + λ 1 − x , β +1 2 (10)
and application of the transformation formula (7) enables one to write (10) in the alternative form: −n, n + λ, λ2 , λ+1 (α,β) (β,α) 2 2 1−x , (11) Rn (x) Rn (x) = 4 F3 λ, α + 1, β + 1 and if we apply Lemma 1 with the following choices p = q = 3, r = s = u = 0, t = 1, [cr ] = [ds ] = [βu ] = φ, z = 1, w = 1 − x 2 , and with a suitable choice of the remaining parameters in relation (6), then the righthand side of relation (11) is turned into n n ν + 21 k λ2 k λ+1 (n + λ)k −n, n + λ, λ2 , λ+1 k 2 k 2 2 1 − x = 4 F3 λ, α + 1, β + 1 (α + 1)k (β + 1)k (k + ν)k (λ)k k=0 , k + ν + 21 k − n, k + n + λ, k + λ2 , k + λ+1 2 1 × 5 F4 k + α + 1, k + β + 1, k + λ, 2k + ν + 1 −k, k + ν 1 − x2 . (12) ×2 F1 ν + 21
Knowing that 2 F1 immediately yield Rn(α,β) (x)
Rn(β,α) (x)
×5 F4
−k, k + ν ν + 21
=
1 − x 2 = C (ν) (x), then relations (11) and (12) 2k
n
n ν + 21 k λ2 k λ+1 k 2 k (n + λ)k
(α + 1)k (β + 1)k (k + ν)k (λ)k 1 k − n, k + n + λ, k + λ2 , k + λ+1 2 , k + ν + 2 1 C (ν) (x), 2k k + α + 1, k + β + 1, k + λ, 2k + ν + 1 k=0
and this completes the proof of Theorem 2.
As an important special case of Theorem 2, and if we take α = β and each is replaced by (α − 21 ), then we get the following important formula: Corollary 1 For every nonnegative integer n and for all α, ν > − 21 , one has (α)k ν + 21 k (n + 2α)k (2α)k α + 21 k (k + ν)k k=0 k − n, k + α, k + n + 2α, k + ν + ×4 F3 k + α + 21 , k + 2α, 2k + ν + 1
n 2 Cn(α) (x) =
123
n k
1 2
1 C (ν) (x). (13) 2k
Author's personal copy Linearization formulae for certain Jacobi polynomials
Remark 1 It is worthy to mention here that relation (13) leads to the same relation obtained by Sanchez [26, formula 18, p. 265], for Gegenbauer polynomials taking into consideration the identity: Cn(α) (x) =
n! ¯ (α) C (x), (2α)n n
(α) where C¯ n (x) is the Gegenbauer polynomial of degree n which used in Ref. [26].
Remark 2 In Dehesa et al. [5, Sects. 3 and 5], the evaluation of entropy and the logarithmic potentials of the Gegenbauer polynomials can be executed by using an expression of the square of such polynomials, and the integral appeared in Eq. (34) can be computed by means of applying formula (13) (for the case ν = 0).
4 Some special linearization formulae As described in this section, and based on using some standard formulae such as Pfaff–Saalschütz identity and also with the application of the algorithms of Zeilberger, Petkovsek and van Hoeij, the hypergeometric function of unit argument that appears in the linearization formula (9) can be reduced into explicit forms for certain choices of the involved parameters, and hence some new special linearization formulae of products of Jacobi polynomials for certain parameters can be deduced. Some of these formulae are considered in the following two subsections.
4.1 Linearization formula for product of Chebyshev polynomials of the third and fourth kinds The linearization formula for the product of Chebyshev polynomials of the third and fourth kinds can be obtained from relation (9), by taking α = − 21 and β = 21 . This result is given in the following corollary: Corollary 2 If we set α = − 21 , β = Vn (x) Wn (x) = (ν)
n k=0
1 2
in the linearization formula (9), then we get
(−1)n+k nk (2k + 1) (n + 1)k ν + 21 k (ν) C2k (x). 3 (k + ν) (2k + ν + 1) (k − n + ν) k n−k 2 k (14)
Proof Substituting α = − 21 , β = (− 21 , 12 )
Rn
1 2
into relation (9), and noting that
(x) = Vn (x),
( 1 ,− 21 )
Rn 2
(x) =
Wn (x) , 2n + 1
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immediately yields Vn (x) Wn (x) = (2n + 1) × 3 F2
n
n k
k=0
(n + 1)k ν + 21 k 3 2 k (k + ν)k
k − n, k + n + 1, k + ν + k + 23 , 2k + ν + 1
1 2
1 C (ν) (x). 2k
(15)
With the aid of the well-known Pfaff–Saalschütz identity (8), the 3 F2 (1) in (15) reduces to 3 F2
k − n, k + n + 1, k + ν + k + 23 , 2k + ν + 1
1 2
n+k (ν) (2k + 1) (2k +ν +1) 1 = (−1) . (2n+1) (k −n+ν) (k +n+ν +1) (16)
Relations (15) and (16) immediately yield Vn (x) Wn (x) = (ν)
n k=0
(−1)n+k nk (2k + 1) (n + 1)k ν + 21 k (ν) C2k (x), 3 (k + ν) (2k + ν + 1) (k − n + ν) k n−k 2 k
and this completes the proof of Corollary 2.
The following three important linearization formulae can be directly deduced from the linearization formula (14), by setting ν = 0, 21 and 1, respectively. Corollary 3 If we set ν = 0, 21 and 1, respectively, in the linearization formula (14), then the following three linearization formulae are obtained: Vn (x) Wn (x) = 1 + 2
n
T2 k (x),
(17)
k=1
Vn (x) Wn (x) =
n (−1)n+k (4k + 1) (k + n)! π P2k (x), 2 (n − k)! k − n + 21 k + n + 23 k=0
Vn (x) Wn (x) = U2n (x).
(18) (19)
Remark 3 It should be noted here that the two linearization formulae (17) and (19) can also follow from their trigonometric representations. 4.2 Some other linearization formulae In this subsection, we give some other linearization formulae of products of Jacobi polynomials for certain parameters. These formulae can be deduced from the linearization formula (9) for ceratin choices of the three parameters α, β and ν. We consider only the following four cases:
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(i) In case of α = − 21 , β = is turned into
3 2
and ν = 1: In such case, the linearization formula (9)
3 2 k n (k + 2)n 2 n = 1 5 (n + 1) 2 n 2 (n + 1)n n 3 3 k − n, k + 1, k + 2 , k + 2 , k + n + 2 (1) ×5 F4 1 C2 k (x). k + 21 , k + 2, k + 25 , 2k + 2
( 3 , −1 ) Rn 2 2 (x)
( −1 , 3 ) Rn 2 2 (x)
(20)
In the following, two methods are derived for the sake of reducing the 5 F4 (1) that appears in formula (20), and hence deduce a new linearization formula. First method: We employ computer algebra. Set Si,n = 5 F4
−i, n − i + 1, n − i + 23 , n − i + 23 , 2n − i + 2 1 , n − i + 21 , n − i + 2, n − i + 25 , 2n − 2i + 2
then by means of Zeilberger’s algorithm (see for instance, Koepf [17] and [18]), via the Maple software, and in particular, sumrecursion command, Si,n , satisfies the following recurrence relation of order two: (i + 1)(i + 2)(i − 2n − 1)(i − 2n)(2i − 2n − 1) Si,n + 4 (i + 2)(i − 2n) ×(2i − 2n − 3)(2i − 2n + 1)(i − n − 1) Si+1,n + 4(2i − 2n − 3) ×(2i − 2n − 1)(2i − 2n + 3)(i − n − 1)(i − n) Si+2,n = 0,
(21)
with the initial values S0,n = 1, S1,n =
4n 2 − 4n − 7 , 2(n + 1)(2n − 1)(2n + 3)
which has the exact solution
Si,n
i! (n − i + 1) (2n − 2i + 3) (2n + 1)2 − 8 i(n + 1) (2n − 2i)! . (22) = (n + 1)(2n + 1)(2n + 3) (2n − i + 1)!
Therefore, the 5 F4 (1) in (20) has the following reduction formula: k − n, k + 1, k + 23 , k + 23 , k + n + 2 5 F4 1 k + 21 , k + 2, k + 25 , 2k + 2 (k + 1)(2k + 3) 8 k n + 8 k − 4n 2 − 4n + 1 (2 k)! (n − k)! = , (n + 1)(2n + 1) (2n + 3)(k + n + 1)!
(23)
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and hence the following linearization formula is obtained: 3 (n + 1)(2n + 3) n 8 k n + 8 k − 4n 2 − 4n + 1 U2 k (x). ×
( 3 ,− 12 )
Rn 2
(− 12 , 23 )
(x) Rn
(x) =
+ 1)2 (2n
(24)
k=0
Remark 4 It is worthy to note here that the exact solution (22) of the recurrence relation (21) can be obtained by means of Petkovsek’s algorithm (see, Koepf [17] and [18, Chapter 9]), or by the improved version of van Hoeij [30]. In addition, the package in Maple called “LREtools[hypergeomsols]” may be used for this purpose. Second method: The idea of this method depends on expressing the 5 F4 (1) in (20) as a combination of some other hypergeometric series with standard reduction formulae. First, we observe that we have the following partial fraction decomposition: (k + 1) j (k + 23 ) j (k + 23 ) j (k + 2) j (k + 25 ) j (k + 21 ) j
=
4(k + 1) (k + 23 ) j (2k + 3) (k + 1) j − . 2k + 1 (k + 25 ) j 2k + 1 (k + 2) j
(25)
Then the 5 F4 (1) in (20) can be written as a linear combination of two terminating one-balanced 3 F2 (1). Explicitly, we can write k − n, k + 1, k + 23 , k + 23 , k + n + 2 1 k + 21 , k + 2, k + 25 , 2k + 2 4(k + 1) k − n, k + n + 2, k + 23 = 3 F2 1 2k + 2, k + 25 2k + 1 (2k + 3) k − n, k + n + 2, k + 1 − 3 F2 1 . 2k + 2, k + 2 2k + 1
5 F4
(26)
If we apply Pfaff–Saalschütz identity (8) twice on (26), then the 5 F4 (1) in (20) can be reduced to the same formula obtained in (23), and hence the linearization formula (24) is obtained. (ii) In case of α = − 21 , β = 23 and ν = 0: In such a case, the linearization formula (9) is turned into n 3 n 3 ( 23 , −1 ( −1 2 k k (n + 2)k 2 ) 2 ,2) Rn (x) Rn (x) = 5 (k)k k=0 (k + 1) 2 k k − n, k + 1, k + 23 , k + n + 2 ×4 F3 1 T2 k (x). k + 2, k + 25 , 2k + 1 Now, if we set G i,n = 4 F3
123
−i, −i + n + 1, −i + n + 23 , −i + 2n + 2 1 , −i + n + 2, −i + n + 25 , −2i + 2n + 1
(27)
Author's personal copy Linearization formulae for certain Jacobi polynomials
then by employing computer algebra, and in particular Zeilberger’s algorithm, G i,n , satisfies the following recurrence relation of order two: 4(2i − 2n − 3)(2i − 2n − 1)(i − n − 2)(i − n − 1)(i − n)(i − n + 1) G i+2,n +4(2i − 2n − 3)(i − n − 2)(i −n)(i −n+1) i 2 −2 i n+2 i − 5 n−3 G i+1,n +(i + 1)(i + 3)(i − 2n − 2)(i − 2n)(i − n − 1)(i − n + 2) G i,n = 0,
(28)
with the initial values G 0,n = 1, G 1,n =
3n + 4 , (n + 2) (2n + 3)
which has the exact solution:
(n−i +1) (i + 1)! (2n − 2i)! (2n − 2i + 3) 4 n 2 + 4 n − 4 i n−4 i +1 , G i,n = (n+1)(2n+1)(2n+3) (2n − i + 1)! (29)
and accordingly k − n, k + 1, k + 23 , k + n + 2 4 F3 1 k + 2, k + 25 , 2k + 1 (k + 1) (2k + 3) (2k)! (n − k + 1)! (4 k n + 4 k + 1) = . (n + 1) (2n + 1) (2n + 3) (k + n + 1)!
(30)
This leads to the following linearization formula: ( 3 ,− 12 )
Rn 2
(− 12 , 23 )
(x) Rn
(x) =
+ 1)2 (2n
6 + 1)(2n + 3)
(n n 1 (n − k + 1)(4 k n + 4 k + 1) T2k (x), × ck
(31)
k=0
where
ck =
2 k = 0, 1, k > 0.
An alternative method to obtain the linearization formula (31): The 4 F3 (1) in (27) can be written with the aid of the partial fraction decomposition as k − n, k + 1, k + 23 , k + n + 2 4 F3 1 k + 2, k + 25 , 2k + 1 k − n, k + n + 2, k + 1 = (2k + 3) 3 F2 1 2k + 1, k + 2 k − n, k + n + 2, k + 23 (32) −2(k + 1) 3 F2 1 . 2k + 1, k + 25
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Now, we use the identity (see, Miller and Paris [22]) m (a) j (b) j j=0
(b + 1)m = (c) j j! m!
3 F2
−m, b, c − a 1 . b + 1, c
(33)
If we choose m = n − k, a = −(n − k + 1), c = 2k + 1, then the left-hand side of (33) equals n−k (−(n − k + 1)) j (b) j −(n − k + 1), b = 2 F1 1 2k + 1 (2k + 1) j j! j=0
−
(−(n − k + 1))n−k+1 (b)n−k+1 . (2k + 1)n−k+1 (n − k + 1)!
Making use of Chu–Vandermonde formula, and performing some manipulations on the right-hand side of (32), enables one to obtain the reduction formula (30). This leads to the same linearization formula as obtained in (31). (iii) In case of α = 23 , β = 21 and ν = 1: In such a case, the linearization formula (9) is turned into (3,1) Rn 2 2 (x)
(1,3) Rn 2 2 (x)
n n 24 k (k + 1)! (k + n + 2)! = (n + 2)! (2k + 4)! k=0 k − n, k + 23 , k + 2, k + n + 3 × 4 F3 1 U2 k (x). k + 25 , k + 3, 2k + 2 (34)
Performing similar procedures to those followed in the two cases (i) and (ii), it can be shown that the 4 F3 (1) in (34) can be reduced to the following form: 4 F3
(k +2) (2k +3) (2k +1)! (n − k +1)! k + 23 , k + 2, k − n, k +n+3 1 = (n+1) (n+2) (2n+3) (k +n+2)! k + 25 , k + 3, 2k +2 × (k(2n + 3) + n + 2),
and this leads to the following linearization formula: ( 3 , 21 )
Rn 2
( 1 , 23 )
(x) Rn 2
(x) =
6 (n + 1)2 (n + 2)2 (2n + 3) n (n − k + 1)(2 k n + 3k + n + 2) U2 k (x). (35) × k=0
(iv) In case of α = 23 , β =
123
1 2
and ν = 0:
Author's personal copy Linearization formulae for certain Jacobi polynomials
In such a case, the linearization formula (9) is turned into (3,1) Rn 2 2 (x)
(1,3) Rn 2 2 (x)
n n 48 k (k + 1)! (k + n + 2)! = (n + 2)! ck (2k + 4)! k=0 k − n, k + 21 , k + 2, k + n + 3 ×4 F3 1 T2 k (x). k + 25 , k + 3, 2k + 1 (36)
It can be shown that the 4 F3 in (36) reduces to k − n, k + 21 , k + 2, k + n + 3 4 F3 1 k + 25 , k + 3, 2k + 1 (k + 2)(2k + 3) (2k + 1)! (n − k + 2)! = 3(n + 1)(n + 2)(2n + 3) (n + k + 2)! × k(2n + 3) + n(n + 3) + 3 ,
and this leads to the following linearization formula: ( 3 , 21 )
Rn 2
( 1 , 23 )
(x) Rn 2
(x) =
+ 1)2 (n
4 + 2)2 (2n + 3)
(n n 1 (k − n − 2)(k − n − 1)(k(2n + 3) × ck k=0
+n(n + 3) + 3) T2 k (x).
(37)
Acknowledgments The authors would like to thank the anonymous referees for their valuable and constructive comments which improved the manuscript in its present form.
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