LINEARIZATION COEFFICIENTS FOR THE JACOBI POLYNOMIALS

Transcription

1 Séminaire Lotharingien de Combinatoire, B16a (1987), 14 pp. [Formerly : Publ. I.R.M.A. Strasbourg, 1987, 341/S 16, p ] LINEARIZATION COEFFICIENTS FOR THE JACOBI POLYNOMIALS BY Dominique FOATA AND Doron ZEILBERGER RÉSUMÉ. Une formule explicite pour les coefficients de linéarisation des polynômes de Jacobi a été donnée par RAHMAN, d où l on tire, sans calcul, les propriétés de positivité. L obtention de la formule de RAHMAN par des méthodes combinatoires semble malaisée. On peut cependant donner plusieurs interprétations combi- natoires de l intégrale du produit de polynômes de Jacobi i P n (α,β) i (x) et en déduire une évaluation dans le cas particulier où n 1 = n + + n m. ABSTRACT. The explicit non-negative representation of the linearization coefficients of the Jacobi polynomials obtained by RAHMAN seems to be difficult to be derived by combinatorial methods. However several combinatorial interpretations can be provided for the integral of the product of Jacobi polynomials i P n (α,β) i (x) and furnish an evaluation of this integral in the particular case where n 1 = n + + n m. 1. Introduction. Standard definition for the Jacobi polynomials reads : P (α,β) n (x) = = n n! n ( )( ) ( ) j ( n + α n + β x 1 x + 1 n j j n ( n j j=0 j=0 ) n j ) (α n j) j (β j) n j (x 1) n j (x + 1) j. (See, e.g., [Er], [Sz]). Let n = (n 1,..., n m ) and consider the integral I n = +1 1 Using the classical evaluation +1 1 it is readily seen that (1 x) α (1 + x) β m (1 x) α (1 + x) β dx = i=1 P n (α,β) i (x) dx. Γ(α + 1)Γ(β + 1), Γ(α + β + ) 73

2 D. FOATA AND D. ZEILBERGER (1.1) I n = with α+β+1 Γ(α + 1)Γ(β + 1) i n L n, i! (α + β + ) Σni Γ(α + β + ) (1.) L n = k ( 1) Σ(n i k i ) (α + 1) Σ(ni k i ) (β + 1) Σki ( ) ni (α n i k i ) ki (β k i ) ni k k i. i i The linearization problem consists of finding an appropriate representation for I n in such a way that non-negative properties of I n are directly apparent from the representation itself. Along those lines RAHMAN [Ra] found the following fantastic formula involving the series 9 F 8 : let s+1 n, 0 j n s and let (1.3) I s+j,n s,n = (α + 1) s+j(α + 1) n s (α + 1) n (s + j)! (n s)! n! Then, for j even Γ(α + 1)Γ(β + 1) (α+β+1) Γ(α + β + 1) (s + j)! (β + 1) s+j g(s + j, n s, n). (α + β) s+j (α + β + 1) s+j (s + j + α + β + 1) g(s + j, n s, n) = α + β s + j (α + β n s) n s α + β + 1 (α + 1) s+j(β + 1) n (α + β + 1) s+j (α + β + 1) j n! (α + 1) s (α + 1) n s (β + 1) s+j (α + β + ) n+j s! j! (s n) j/ (α + β + n + 1) ( j/ s n α + β ) (s + 1) j/ (α + 1) j/ j/ (s n α) j/ (β + n + 1) j/ (1/) j/ ( 1 + s n α + β ) (s + 1) j/ (α + 1) j/ j/ [ α α, F, α + 1, α β 8 α, 1, α + β α β + 1, + 1, α + β + 1, α + β + n j, s n + j, s j, j ] β n j, α + n + 1 s j, α + s j, α j 74

3 JACOBI POLYNOMIALS and a corresponding formula for j odd. Note that in RAHMAN s paper [Ra, p. 917, formula (1.7)] the factor (α + β n s) n s occurring after the first fraction above is missing. As proved by RAHMAN [Ra], the foregoing formula shows that if j, s, n are non-negative integers with s + 1 n, 0 j n s, then g(s + j, n s, n) 0 whenever α β > 1 and α + β Putting back the value of g(s + j, n s, n) into (1.3) we deduce the following formula L s+j,n s,n = (α + 1) n(s + j)! (α + 1) s+j (β + 1) n (α + β + 1) s+j (α + β + 1) s+j (α + 1) s (α + β n s) n s(α + β + 1) j n! s! j! (s n) j/ (α + β + n + 1) ( j/ s n α + β ) (α + s + 1) j/ j/ (s n α) j/ (β + n + 1) j/ (1/) j/ ( 1 + s n α + β ) (s + 1) j/ (α + 1) j/ j/ [ α α, F, α + 1, α β 8 α, 1, α + β α β + 1, α + β + n j, + 1, α + β + 1, β n j, s n + j, s j, j α + n + 1 s j, α + s j, α j when j is even. When j is odd, formula (1.8) of RAHMAN [Ra] leads to : L s+j,n s,n = (α + 1) n(s + j)! (α + 1) s+j (β + 1) n (α + β + 1) s+j (α + 1) s (α + β + 1) s+j(α + β n s) n s (α + β + 1) j n! s! j! ], (s n) (j+1)/ (α + β + n + 1) ( (j+1)/ s n α + β ) (α + s + 1) (j+1)/ (j+1)/ (s n α) (j 1)/ (β + n + 1) (j 1)/ (3/) ( (j 1)/ 1 + s n α + β ) (s + 1) (j 1)j/ (α + ) (j 1)/ (j 1)/ 75

4 α β α + β F 8 1 j D. FOATA AND D. ZEILBERGER [ α + 3 α + 1,, α + 1, α β α + 1, 3, α + β + 1, α β + 1, + 1, α + β + 3, α + β + n j, s n j, 1 s j, 1 j β n, α + n + 3 s j, α + s j, α j It seems that a derivation of RAHMAN s formula by means of combinatorial methods is out of scope. It would first require an interpretation of the factor (α) k (1 + α/) k (α + 1/) k /(α/) k (1/) k occurring in the series 9 F 8. But such a factor already occurs in each classical hypergeometric series identity involving p F p+1 for p 3, for instance in the Dougall, Whipple and Bailey identities (see [Bai, Chap. 4]). When α = β (the case of ultraspheric polynomials), the factor 9 F 8 vanishes and RAHMAN s formula greatly simplifies. For instance, for j even we get : if 0 j n s L s+j,n s,n = ( ) s + j s! s, j/, j/ if n s j n s ( ) s + j L s+j,n s,n = s! s, j/, j/ In particular, with j = 0 we get ( ) ( ) j n! (n s)! s + j/ (α j/) n j/ (α s + j/) j/ (α n s j/) j/ (β + 1) n+j/ (α + β s + j) s (α + β n s) n s j (α + β + 1) j (α + β + n + 1) j/ ; ( ) ( ) j n! (n s)! s + j/ (α j/) n j/ (α s + j/) j/ (α n s j/) j/ (β + 1) n+j/ (α + β s + j) s (α + β n s) n s (α + β + 1) n s j (α + β + n + 1) j/. (1.5) L s,n s,n = (α +β +s+1) s (α +β +1+n s) n s n! (α +1) n (β +1) n, a formula that will be extended further in the paper. 76 ].

5 JACOBI POLYNOMIALS The purpose of this article is to give a combinatorial interpretation to L n and to deduce from it several analytic consequences (sections and 3). As mentioned previously, we cannot derive RAHMAN s formula, but we can, at least, evaluate an extension of (1.5), that is, (1.6) L n = (α+β +n +1) n (α+β +n m +1) nm n 1! (α+1) n1 (β +1) n1, when m is arbitrary and n 1 = n + +n m. This is presented in section 4.. Weighted bipermutations. Consider formulas (1.1) and (1.). The expression L n will first be proved to be the generating function for certain combinatorial objects, called weighted bipermutations, as follows. Let N = N N m be an ordered partition of a set N with N i = n i (i = 1,,..., m). If is a subset of N, let k i = N i and i = k i (i = 1,,..., m). Next consider a permutation π of the set N. An element x of N is said to be π-incestuous, if both x and π(x) belong to the same component N i. Denote by Inc π the set of all π-incestuous elements of N. Finally, define a weighted bipermutation of N = N N m as being a triple (π 1, π, ), where π 1 and π are permutations of N and is a subset of N that satisfies the properties : Inc π 1 and N \ Inc π. Define the weights of a weighted bipermutation (π 1, π, ) to be : w(α, β; π 1, π, ) = (α + 1) cyc π 1 (β + 1) cyc π ; w (α, β; π 1, π, ) = ( 1) N\ (α + 1) cyc π 1 (β + 1) cyc π ; where cyc π designates the number of cycles of the permutation π. Theorem 1. The polynomial L n defined in (1.) is the generating function for the weighted bipermutations by the weight w. In other words, L n (α, β) := L n = w (α, β; π 1, π, ) = ( 1) N\ (α + 1) cyc π 1 (β + 1) cyc π. Proof. Let (π 1, π, ) be a weighted bipermutation of N = N N m. To the pair (π 1, ) we can associate a sequence (π 11,..., π 1m, σ 1 ), where each π 1i is an injection of i into N i (i = 1,,..., m) and σ 1 is a permutation of the set N \ = i (N i \ i ). Moreover, cyc π 1 = i cyc π 1i + cyc σ 1 and the mapping (π 1, ) (π i1,..., π im, σ 1 ) is bijective. Such a mapping has been described in [Fo-Ze]. Fig. 1 indicates the construction of such a bijection In the same manner, we associate a 77

6 D. FOATA AND D. ZEILBERGER N \ N \ 1 N 1 \ 1 N \ 3 N 3 \ 3 π 1 π 11, π 1, π 13 σ 1 Fig. 1 sequence (π 1,..., π m, σ ) to π, where each π i is an injection of N i \ i into N i (i = 1,,..., m) and σ is a permutation of. Therefore, (α+1) Σ(ni k i ) i (α+1+n i k i ) ki is the generating function for permutations π 1 by number of cycles satisfying Inc π 1. In the same manner, (β + 1) Σki i (β k i) ni k i is the generating function for permutations π by number of cycles satisfying N \ Inc π. Thus, to calculate L n we can first fix k, then a sequence = ( 1,..., m ) with i = k i (i = 1,,..., m) and finally sum over all weighted bipermutations (π 1, π, ). As an application of this combinatorial interpretation we can state the following corollary and also obtain another combinatorial interpretation in terms of pairs of permutations with prescribed incestuous element sets. Corollary 1. If N = n, then L n (β, α) = ( 1) N L n (α, β). In particular, when n is odd, L n (α, α) = 0. Proof. Consider the transformation (π 1, π, ) (π, π 1, N \ ). Then w (β, α; π, π 1, N \ ) = ( 1) (β + 1) cyc π (α + 1) cyc π 1 Thus L n (β, α) = ( 1) N L n (α, β). = ( 1) N ( 1) N\ (α + 1) cyc π 1 (β + 1) cyc π = ( 1) N w (α, β; π 1, π, ). 78

7 JACOBI POLYNOMIALS Corollary (Second combinatorial interpretation). One has : L n = ( 1) N\ (α + 1) cyc π 1 (β + 1) cyc π, where the summation is over all triples (π 1, π, ) with π 1 and π permutations of N, and a subset of N with the property that = Inc π 1 and N \ = Inc π. Proof. Let (π 1, π, ) be a weighted bipermutation. If Inc π 1 Inc π is non-empty, look at the smallest element ξ in that set. Then define φ(π 1, π, ) = (π 1, π, \{ξ}) or (π 1, π, +{ξ}), depending on whether ξ is in or not. In both cases φ(π 1, π, ) is a weighted bipermutation and w φ(α, β; π 1, π, ) = w (α, β; π 1, π, ). Therefore the summation w (α, β; π 1, π, ) over all pairs (π 1, π ) such that Inc π 1 Inc π = equals 0. Now as Inc π 1 and N \ Inc π, the condition Inc π 1 Inc π = means that = Inc π 1 and N \ = Inc π. 3. Weighted derangements. The polynomial L n can also be expressed in terms of derangement polynomials as follows. eep the same notations as in the beginning of section for N,, N i, i and define a -derangement to be a permutation σ of such that for every x in the elements x and σ(x) belong to different components i and j (i j). Set D(k; α) = (α + 1) cyc σ, where σ ranges over all -derangements. Theorem 3 (third combinatorial interpretation). One has : (3.1) L n = N( 1) \N D(n k; α)d(k; β) i (α n i k i ) ki (β k i ) ni k i. Proof. Consider a weighted bipermutation (π 1, π, ) with = Inc π 1 and N \ = Inc π. If we use the bijection described in the proof of THEOREM 1, the pair (π 1, ) is transformed into a sequence 79

8 D. FOATA AND D. ZEILBERGER N \ 1 N 1 \ 1 N \ 3 N 3 \ 3 π 1 N \ π 11, π 1, π 13 σ 1 Fig. (π 11,..., π 1m, σ 1 ), but this time σ 1 is an (N \ )-derangement, as shown in Fig.. In the same way, (π, N \) is transformed into (π 1,..., π m, σ ) with σ being a -derangement. Therefore L n = ( 1) ( ) N\ (α + 1) cyc π 1i (β + 1) cyc π i N i (α + 1) cyc σ 1 (β + 1) cyc σ. As π 1i (resp. π i ) is an injection of i (resp. N i \ i ) into N i and σ 1 (resp. σ ) is an (N \ )-derangement (resp. a -derangement), the summation over the π 1i s, the π i s and the derangements σ 1 and σ yields (3.1). There are several consequences of this interpretation when n 1 n + + n m. First, study the case of the strict inequality. Lemma 1. If k 1 > k + + k m, then D(k, α) = 0. Proof. If π is a -derangement, then π( 1 ) + + m. But i = k 1 > k + + k m and there do not exist any -derangements under this hypothesis. Lemma. Suppose n 1 > n + + n m and let 0 k i n i for i = 1,..., m. Then either k 1 > k + + k m, or (n 1 k 1 ) > (n k ) + + (n m k m ). 80

9 JACOBI POLYNOMIALS Proof. If k 1 k + + k m, then n 1 k 1 > n + + n m k 1 > n k + + n m k m. Proposition 1. If n 1 > n + + n m, then L n = 0. Proof. With the foregoing hypothesis, either k 1 > k + + k m or (n 1 k 1 ) > (n k ) + + (n m k m ). Then, either D(k; β) = 0, or D(n k; α) = 0. Therefore, (3.1) shows that L n = 0. Corollary. If n 1 n, then L n1,n = 0. This is precisely the orthogonality relation. 4. The evaluation of L n for n 1 = n + + n m. Consider again the summation (3.1). When n 1 = n + + n m, the inequality k 1 < k + + k m implies n 1 k 1 > (n k ) + + (n m k m ). Therefore, the factor D(n k; α) vanishes for such a sequence k. In the same manner, if k 1 > k + + k m, then D(k; β) = 0. The summation (3.1) can then be restricted to those sequences k satisfying (4.1) 0 k 1 = k + + k m n 1 = n + + n m. In particular, for m = and n 1 = n = n we obtain : n (4.) L n,n = D(n k, n k, α)d(k, k, β) k=0 (( ) ) n (α n k) k (β k) n k. k We will have a more precise evaluation further in the paper. For the time being, let us compare (4.) with the classical evaluation of the integral : I n,n = +1 1 (1 x) α (1 + x) β( P (α,β) n (x)) dx = 1+α+β Γ(1 + α + n)γ(1 + β + n) n! (1 + α + β + n)γ(1 + α + β + n). (See, e.g., [Rai, p. 60 (11)].) By comparison with the definition of L n,n (formula (1.)), so that (4.3) I n,n = 1+α+β Γ(α + 1)Γ(β + 1) L n,n, n! n! (α + β + ) n Γ(α + β + ) L n,n = (α + β + n + 1) n n! (α + 1) n (β + 1) n. 81

10 D. FOATA AND D. ZEILBERGER This formula will be a consequence of the next theorem. Theorem 4. When n 1 = n + + n m, then L n = (α + β + n + 1) n (α + β + n m + 1) nm n 1! (α + 1) n1 (β + 1) n1. Proof. As just noticed, the summation (3.1) can be restricted to the sequences k satisfying (4.1). But if (π 1, π, ) is a triple with 1 = + + m, then N \ = N 1 + +N m \( m ) = N 1 \ N m \ m = N 1 \ 1, so that the sign ( 1) N\ is always equal to 1. Therefore, L n = (α + 1) cyc π 1 (β + 1) cyc π, where π 1 and π are permutations of N and π 1 ( i ) N i, π 1 (N i \ i ) N \ N i, π (N i \ i ) N i, π ( i ) N \ N i for i = 1,,..., m. Let M 1 = 1 + j (N j \ j ) and M = N 1 \ 1 + j j. Note that M 1 = M = N 1. Our purpose is now to construct a bijection that will explain the occurrence of each factor in the right-hand side of the THEOREM 4 formula. The reader is advised to follow the construction by looking at the geometric representations of the mappings in Fig. 3 (i) For each i =,..., m let f i be the mapping of N i into itself defined by : f i i = π 1 i and f i Ni \ i = π Ni \ i. As π 1 ( i ) N i and π (N i \ i ) N i, the pair ( i, f i ) is a so-called Jacobi endofunction (see [Fo-Le]). Denote by a(f i ) (resp. b(f i )) the number of cycles of f i all vertices of which are in i (resp. (N i \ i ) and let the weight of ( i, f i ) be defined by As shown in [Fo-Le, théorème 1] w( i, f i ) = (α + 1) a(f i) (β + 1) b(f i). (4.4) w(i, f i ) = (α + β + n i + 1) ni, the summation being over all Jacobi endofunctions on N i. (ii) Consider x N 1 \ 1. As π 1 (N 1 \ 1 ) N + + N m and π 1 ( i ) N i (i =,..., m), sooner or later the iterates π1 k (x) will hit the set j (N j \ j ). Let k(x) be the smallest integer satisfying π k(x) 1 (x) j (N j \ j ) and define γ(x) = π k(x) 1 (x). Clearly γ : N 1 \ 1 j (N j \ j ) 8

11 JACOBI POLYNOMIALS N \ N 1 N N 3 N \ 7 π 1 N \ N 1 N N 3 π N \ f and f 3 N \ 6 N 1 N δ and γ N \ N 3 σ 1 Fig. 3 σ 83

12 D. FOATA AND D. ZEILBERGER is a bijection. In the same manner, define a bijection δ : 1 j j by means of π. (iii) With the pair (γ, δ) we can then make up a bijection of N 1 onto the union j N j. (iv) Consider the cycles of π 1. All the cycles that intersect M 1 but sometimes leave M 1 will be made purely M 1 by cancelling all the portions that leave M 1. This gives a permutation σ 1 : M 1 M 1. (v) In the same manner, obtain a permutation σ : M M. Remembering the definitions of a(f i ) and b(f i ) given in (i) we see that to each triple (π 1, π, ) there corresponds a sequence with Accordingly, (, f,..., m, f m, γ, δ, σ 1, σ ) cyc π 1 = a(f ) + + a(f m ) + cyc σ 1 ; cyc π = b(f ) + + b(f m ) + cyc σ. w(α, β; π 1, π, ) = w(, f ) w( m, f m )(α + 1) cyc σ 1 (β + 1) cyc σ. It can be verified that the correspondence between triples and sequences defined by (i) (v) is one-to-one. Hence w(α, β; π1, π, ) = w(, f ) w( m, f m ) γ,δ 1 (α + 1) cyc σ 1 σ 1 σ (β + 1) cyc σ = (α + β + n + 1) n (α + β + n m + 1) nm n 1! (α + 1) n1 (β + 1) n1. Remark. identity (4.3). Note that for m = and n 1 = n = n THEOREM 4 yields 5. Concluding remarks. The problem of the linearization of the classical orthogonal polynomials has been studied again recently by means of combinatorial methods. ASEY and his coauthors [As-Is, As-Is-o] have already obtained several significant results concerning the Hermite, Laguerre and Meixner polynomials. AZOR, GILLIS and VICTOR [Az-Gi-Vi] found an elegant set-up for the Hermite polynomials. The two authors of the present paper [Fo-Ze] have completed the work of ASEY and his coauthors as far as the (generalized) Laguerre polynomials are concerned by exploiting a β-extension of the MacMahon Master Theorem. ZENG [Ze] has further extended the works of ASEY and the two authors and 84

13 JACOBI POLYNOMIALS proved new positivity results concerning the Meixner, rawtchouk and Charlier polynomials. GILLIS and his coauthors [Gi-Je-Ze] reproved several positivity results on the Legendre polynomials. Some of their arguments have been implicitly used in the present paper. RAHMAN s formula, as said in the introduction, should discourage several researchers. Formula (3.1) seems to indicate that a new algebraic tool has to be found to handle the product of two derangement polynomials. There is also a linearization coefficient formula for the ultraspheric polynomials found by HSÜ [Hs], more compact than RAHMAN s formula for α = β, but not so easy to be tackled by combinatorial methods. The only hope for the time being seems to be the symmetric function approach due to ZENG. He has already got an explicit formula for a single derangement polynomial that led to the positivity result for the rawtchouk polynomials. Note added in The article [Ze] has been updated in the reference list. Notice that the linearization coefficients for the class of Sheffer polynomials has been thoroughly studied by Zeng [Ze9]. An interesting connection between linearization coefficeints for Jacobi polynomials and permutation pair counting has been derived in [Ze91]. The Rahman formula remains untamed in the combinatorial environment. [Ze91] ZENG (Jiang). Counting a pair of permutations and the linearization coefficients for Jacobi polynomials, Actes de l Atelier de Combinatoire francoquébecois, Publ. LACIM, no. 10, Université du Québec à Montréal, [Ze9] ZENG (Jiang). Weighted derangements and Sheffer polynomials, Proc. London Math. Soc., t. 65, 199, p

14 D. FOATA AND D. ZEILBERGER REFERENCES [As-Is] ASEY (Richard) and ISMAIL (Mourad E.H.). Permutation problems and special functions, Canad. J. Math., t. 8, 1976, p [As-Is-o] ASEY (Richard), ISMAIL (Mourad E.H.) and OORNWINDER (T.). Weighted permutation problems and Laguerre polynomials, J. Combinatorial Theory, Ser. A, t. 5, 1978, p [Az-Gi-Vi] AZOR (R.), GILLIS (J.) and VICTOR (J.D.). Combinatorial application of Hermite polynomials, SIAM J. Math. Anal., t. 13, 198, p [Bai] BAILEY (W.N.). Generalized Hypergeometric Series. Cambridge Univ. Press, [Er] ERDÉLYI (A.) et al. Higher Transcendental Functions (Bateman Manuscript Project), 3 vol. New York, McGraw-Hill, [Fo-Le] FOATA (Dominique) et LEROUX (Pierre). Polynômes de Jacobi, interprétation combinatoire et fonction génératrice, Proc. Amer. Math. Soc., t. 87, 1983, p [Fo-Ze] FOATA (Dominique) and ZEILBERGER (Doron). Weighted derangements and Laguerre polynomials, in Séminaire Lotharingien de combinatoire (Bayreuth, Erlangen, Strasbourg), 8 e session [D. Foata, éd. ; Sainte-Croix-aux-Mines. 3 8 mai 1983], p Publ. IRMA Strasbourg, 9/S 08, [Gi-Je-Ze] GILLIS (J.), JEDWAB (J.) and ZEILBERGER (D.). A combinatorial interpretation of the integral of the product of Legendre polynomials, manuscrit, Weizmann Inst., [Hs] HSÜ (Hsien-Yü). Certain integrals and infinite series involving ultraspheric polynomials and Bessel functions, Duke Math. J., t. 4, 1938, p [Ra] RAHMAN (Mizan). A non-negative representation of the linearization coefficients of the product of Jacobi polynomials, Canad. J. Math., t. 33, 1981, p [Rai] RAINVILLE (E.D.). Special Functions. New York, Chelsea, [Sz] SZEGÖ (Gabor). Orthogonal Polynomials. Providence, R.I., Amer. Math. Soc., 1939 (Amer. Math. Soc. Colloq. Publ., 3) [second printing of the fourth edition : 1978]. [Ze] ZENG (Jiang). Linéarisation de produits de polynômes de Meixner, rawtchouk et Charlier, S.I.A.M. J. Math. Anal., t. 1, 1989, p Dominique Foata, Département de mathématique, Université Louis-Pasteur, 7, rue René-Descartes, F Strasbourg, France. Doron Zeilberger, Department of Mathematics, Drexel University, Philadelphia, Pennsylvania 19104, U.S.A. now at Temple University, Philadelphia, Pennsylvania 191, U.S.A. 86