- Email: [email protected]
Orthogonality of some polynomial sets via quasi-monomiality
Applied Mathematics and Computation 141 (2003) 415–425 www.elsevier.com/locate/amc
Orthogonality of some polynomial sets via quasi-monomiality H.M. Srivastava
a,*
, Y. Ben Cheikh
b
a
b
Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3P4, Canada D epartement de Math ematiques, Facult e des Sciences, Universit e du Centre, TN-5019 Monastir, Tunisia
Abstract The lowering operator r and the raising operator s, associated with a polynomial set fPn g1 n¼0 , are two operators which are independent of n and satisfy the relationships rðPn ÞðxÞ ¼ nPn1 ðxÞ
and sðPn ÞðxÞ ¼ Pnþ1 ðxÞ ðn 2 N0 Þ:
In this paper, we use these operators to study the orthogonality of some polynomial sets. More precisely, we state sufficient conditions, in terms of the r and s operators, to ensure the orthogonality. We also express explicitly, by means of the r operator, the linear functional for which the orthogonality holds true. We obtain some well-known results as particular cases of the results presented in this paper. Ó 2003 Elsevier Science Inc. All rights reserved. Keywords: Orthogonal polynomials; Lowering operators; Raising operators; Linear functionals; Favard theorem; Boas–Buck polynomials; Appell polynomials; Generating functions; Charlier polynomials; Hermite polynomials; Laguerre polynomials; Meixner polynomials
1. Introduction Let P be the vector space of polynomials with coefficients in C. A polynomial sequence fPn g1 n¼0 in P is called a polynomial set if and only if *
Corresponding author. E-mail addresses: [email protected], [email protected] (Y. Ben Cheikh).
[email protected]
(H.M.
0096-3003/03/$ - see front matter Ó 2003 Elsevier Science Inc. All rights reserved. doi:10.1016/S0096-3003(02)00961-X
Srivastava),
416
H.M. Srivastava, Y. Ben Cheikh / Appl. Math. Comput. 141 (2003) 415–425
degðPn Þ ¼ n
ðn 2 N0 :¼ f0; 1; 2; . . .gÞ:
For an integer j, we denote by KðjÞ the space of operators acting on analytic functions that augment (resp. reduce) the degree of every polynomial by ex actly j if j 2 N0 (resp. j 2 Z 0 ), Z0 being the set of nonpositive integers. This tacitly includes the fact that, if r 2 Kð1Þ , then rð1Þ ¼ 0. In an earlier paper [1], it was shown that every polynomial set fPn g1 n¼0 is quasi-monomial. This means that there exists a lowering operator r and a raising operator s, independent of n, such that rðPn ÞðxÞ ¼ nPn1 ðxÞ
and
sðPn ÞðxÞ ¼ Pnþ1 ðxÞ ðn 2 N0 Þ;
ð1Þ
where r and s are in Kð1Þ and Kð1Þ , respectively. Most of the properties of polynomial sets can be deduced by using operational rules with r and s operators. In many works (see, for instance, [5–10]), new and known results related to Hermite polynomials, Laguerre polynomials, Legendre polynomials, and Appell polynomials were derived by using the monomiality principle. The main properties considered in these recent works are related to generating functions, differential equations, and Rodrigues formulas. In this paper, we study the orthogonality of some polynomial sets by using the corresponding lowering and raising operators. In Section 2 of this paper, we recall some useful definitions and results. Then, in Section 3, we present our main result (Theorem 3.1) from which we deduce well-known properties related to the orthogonality of some classical orthogonal polynomials.
2. Basic definitions and results Let P0 be the dual of the vector space P of polynomials with coefficients in C. We denote by hu; f i the effect of the linear functional u 2 P0 on the polynomial f 2 P. Let fPn g1 n¼0 be a polynomial set in P. The corresponding monic 1 b polynomial sequence f P n g is given by n¼0
Pn ¼ kn Pbn
ðn 2 N0 Þ;
where kn is the normalization coefficient and its dual sequence f Pen g1 n¼0 is defined by h Pen ; Pbm i ¼ dn;m
ðn; m 2 N0 Þ;
dn;m being the Kronecker delta. 1 The polynomial sequence fPn gn¼0 is called an orthogonal polynomial set with respect to the functional L if it satisfies the following condition: hL; Pm Pn i ¼ cn dn;m
ðm; n 2 N0 Þ:
ð2Þ
H.M. Srivastava, Y. Ben Cheikh / Appl. Math. Comput. 141 (2003) 415–425
417
The so-called Favard theorem asserts that the orthogonality condition (2) may 1 be deduced from the fact that the sequence f Pbn gn¼0 satisfies a second-order (three-term) recurrence relation [4]: Pbnþ2 ðxÞ ¼ ðx bnþ1 Þ Pbnþ1 ðxÞ cnþ1 Pbn ðxÞ ðn 2 N0 Þ ð3Þ Pb0 ðxÞ ¼ 1 and Pb1 ðxÞ ¼ x b0 and the regularity conditions: ð4Þ cnþ1 6¼ 0 ðn 2 N0 Þ: Sufficient conditions, in terms of lowering and raising operators, to ensure the orthogonality of a polynomial set are given by 1
Lemma 2.1. Let fPn gn¼0 be a polynomial set and let r and s be its lowering and raising operators. If 1 s ¼ a1 þ ða2 þ a3 xÞr þ a4 r2 ða5 þ a6 x þ a7 rÞ; ð5Þ where a1 ; . . . ; a7 are complex parameters constrained by a1 6¼ 0;
ða3 ; a6 Þ 6¼ ð0; 0Þ;
and
ða4 ; a7 Þ 6¼ ð0; 0Þ;
ð6Þ
then the conditions (3) and (4) are satisfied. 1
A polynomial set fBn gn¼0 is called the sequence of basic polynomials for r 2 Kð1Þ if (i) B0 ðxÞ ¼ 1, (ii) Bn ð0Þ ¼ 0 whenever n 2 N :¼ N0 n f0g, (iii) rðBn ÞðxÞ ¼ nBn1 ðxÞ ðn 2 N0 Þ. It is known that every r 2 Kð1Þ has a sequence of basic polynomials, and we have 1
Lemma 2.2 (cf. [2], Lemma 2.2). Let fPn gn¼0 be a polynomial set and let r be its 1 lowering operator. Also let fBn gn¼0 be the sequence of basic polynomials for r. Then there exists a unique power series 1 X uðtÞ ¼ ak tk ða0 6¼ 0Þ k¼0
such that uðrÞðPn ÞðxÞ ¼ Bn ðxÞ
ðn 2 N0 Þ:
We say that fPn g1 n¼0 is a r-Appell polynomial set of the transfer power series AðtÞ, where 1 : AðtÞ ¼ uðtÞ
418
H.M. Srivastava, Y. Ben Cheikh / Appl. Math. Comput. 141 (2003) 415–425
According to Lemma 2.2 and another known result [1, Theorem 2.1], every polynomial set can be viewed as a r-Appell polynomial set of the transfer power series AðtÞ for suitable r and AðtÞ, which may be deduced from the following result: 1
Lemma 2.3. A r-polynomial set fPn gn¼0 of the transfer power series AðtÞ is generated by Gðx; tÞ ¼ AðtÞG0 ðx; tÞ ¼
1 X n¼0
Pn ðxÞ
tn ; n!
where G0 ðx; tÞ is a solution of the system: rG0 ðx; tÞ ¼ tG0 ðx; tÞ G0 ðx; 0Þ ¼ 1; and conversely. An explicit expression of the dual sequence of a r-polynomial set fPn g1 n¼0 of the transfer power series AðtÞ was given in [2] by 1 rn rn e ðf ÞðxÞ f ð0Þ ðn 2 N0 ; f 2 PÞ: ¼ ð7Þ hPn; f i ¼ n! AðrÞ n!AðrÞ x¼0 In our present investigation, we need also the following two lemmas (see also [3]). 1
Lemma 2.4 (cf. [1], Corollary 3.3). Let P :¼ fPn gn¼0 be a Boas–Buck polynomial set generated by 1
X tn Gðx; tÞ ¼ AðtÞB xCðtÞ ¼ Pn ðxÞ ; n! n¼0
ð8Þ
where AðtÞ ¼ BðtÞ ¼
1 X k¼0 1 X
ak tk ða0 6¼ 0Þ;
CðtÞ ¼
1 X
ck tkþ1 ðc0 6¼ 0Þ;
and
k¼0
bk tk ðbk 6¼ 0; k 2 N0 Þ:
ð9Þ
k¼0
Also let q 2 Kð1Þ be given by qð1Þ ¼ 0
and
qðxn Þ ¼
bn1 n1 x bn
ðn 2 NÞ
ð10Þ
H.M. Srivastava, Y. Ben Cheikh / Appl. Math. Comput. 141 (2003) 415–425
419
and suppose that q1 ðxn Þ ¼
bnþ1 nþ1 x bn
ðn 2 N0 Þ: 1
Then the polynomial set fPn gn¼0 is quasi-monomial under the action of A0 ðrÞ d 0 1 0 þ xDx C ðrÞq r ¼ C ðqÞ and s ¼ A ðtÞ ¼ Dt AðtÞ; Dz :¼ ; AðrÞ dz ð11Þ where C is the inverse of C, that is,
C CðtÞ ¼ C C ðtÞ ¼ t
with
C ðtÞ ¼
1 X
cn tnþ1
ðc0 6¼ 0Þ:
ck tkþ1
ðc0 6¼ 0Þ
n¼0
Lemma 2.5 (cf. [1], Corollary 3.4). Let AðtÞ ¼
1 X
ak t k
ða0 6¼ 0Þ
and
CðtÞ ¼
k¼0
1 X
ð12Þ
k¼0
be two analytic functions. 1 Then the polynomial set Q :¼ fQn gn¼0 defined the generating function: 1 x=x X tn AðtÞ 1þ xCðtÞ ¼ Qn ðx; xÞ n! n¼0
ð13Þ
is quasi-monomial under the action of r ¼ C ðDx Þ and
s¼
A0 ðrÞ þx AðrÞ
C 0 ðrÞ A0 ðrÞ þ xC 0 ðrÞTx ; ¼ 1 þ xDx AðrÞ
ð14Þ
where Tj is the translation operator and Dx ¼
Tx 1 : x
ð15Þ
3. Orthogonality of a class of polynomial sets Our approach for studying the orthogonality of a polynomial set fPn g1 n¼0 consists of two steps. Firstly, in order to assert the orthogonality of the considered polynomial set, we use the Favard theorem via Lemma 2.1. We then
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H.M. Srivastava, Y. Ben Cheikh / Appl. Math. Comput. 141 (2003) 415–425
take L ¼ Pe0 as the linear functional, for which we have the orthogonality, and we express it by means of (7). We thus obtain our main result given by 1
Theorem 3.1. Let fPn gn¼0 be a r-polynomial set of the transfer power series AðtÞ and let s be its raising operator. If s is given by (5) under the constraints (6), then 1 fPn gn¼0 is an orthogonal polynomial set with respect to the linear functional L given by hL; f i ¼
1 ðf Þð0Þ ðf 2 PÞ: AðrÞ
ð16Þ
We now apply our main result (Theorem 3.1) to some well-known orthogonal polynomial sets. 3.1. Charlier polynomials The monic Charlier polynomials CðaÞ n ðxÞ are generated by ð1 þ tÞx expðatÞ ¼
1 X
CnðaÞ ðxÞ
n¼0
tn n!
ða 6¼ 0; jtj < 1Þ:
ð17Þ
1
It follows then that the polynomial set fCðaÞ n gn¼0 is a D-Appell polynomial set of the transfer power series AðtÞ ¼ eat , where, for convenience, D :¼ D1 . According to Lemma 2.5, fCnðaÞ g1 n¼0 is quasi-monomial under the action of r¼D
s ¼ a þ xT1 :
and
The raising operator s may be rewritten as follows: 1
s ¼ ð1 þ DÞ ð1 a þ x aDÞ; since T1 ¼ 1 þ D
and
T1 x xT1 ¼ T1 :
Then the conditions (5) and (6) are satisfied and the orthogonality of the Charlier polynomials follows. The linear functional L for which we have this orthogonality can be deduced from (16), and we thus obtain hL; f i ¼ expðaDÞðf Þð0Þ ¼
1 X ak k D ðf Þð0Þ k! k¼0
which, in view of the well-known relationship: k X k Dk f ð0Þ ¼ f ðjÞ; ð1Þkþj j j¼0 assumes the form:
ðf 2 PÞ;
ð18Þ
H.M. Srivastava, Y. Ben Cheikh / Appl. Math. Comput. 141 (2003) 415–425
hL; f i ¼
1 X ak k! k¼0
k X
ð 1Þkþj
j¼0
k f ðjÞ j
! ¼
421
1 X 1 X akþj f ðjÞ ð1Þk k!j! k¼0 j¼0
1 X aj ¼ ea f ðjÞ: j! j¼0
ð19Þ
3.2. Hermite polynomials The classical Hermite polynomials Hn ðxÞ are generated by 1 X tn ðt 2 CÞ: Hn ðxÞ expð2xt t2 Þ ¼ n! n¼0
ð20Þ
1
It follows then that the polynomial set fHn gn¼0 is an Appell polynomial set (cf., e.g., [11]) of the transfer power series AðtÞ ¼ expðt2 Þ: According to Lemma 2.4, fHn g1 n¼0 is quasi-monomial under the action of 1 ð21Þ r ¼ Dx and s ¼ 2x Dx ¼ 2x 2r: 2 The expression for the raising operator s in (21) asserts the orthogonality of the Hermite polynomials. The linear functional L for which we have this orthogonality may be deduced from (16) as follows: 1 X 1 D2k x ðf Þð0Þ hL; f i ¼ expðr2 Þðf Þð0Þ ¼ exp D2x ðf Þð0Þ ¼ ðf 2 PÞ: 4 4k k! k¼0 ð22Þ n
In particular, for f ðxÞ ¼ x (n 2 N0 ), we have hL; xn i 8 <0 Z 1 n! 1 nþ1 2 ¼
p ffiffiffi p ffiffiffi ¼ xn expðx2 Þ dx C ¼ : n n 2 p p 0 2 2 ! Z 1 1 p ffiffiffi ¼ xn expðx2 Þ dx: p 1
if n is odd if n is even
ð23Þ 3.3. Laguerre polynomials The Laguerre polynomials LðaÞ n ðxÞ are generated by 1 xt X a1 n exp LðaÞ ðjtj < 1Þ; ð1 tÞ ¼ n ðxÞt 1t n¼0
ð24Þ
422
H.M. Srivastava, Y. Ben Cheikh / Appl. Math. Comput. 141 (2003) 415–425
which is of the type (8) in Lemma 2.4 with AðtÞ ¼ ð1 tÞa1
CðtÞ ¼ C ðtÞ ¼
and
t : 1t
1
It follows then that the polynomial set fn!LnðaÞ gn¼0 is quasi-monomial under the action of r¼
Dx 1 Dx
s¼
and
aþ1 x 1r
1 ð1 rÞ
2
:
Using the fact that 1 Dx ¼
1 1r
and
Dx x xDx ¼ 1;
we find that 1 ð1 rÞ
2
xx
1 2
ð1 rÞ
¼
2 ; 1r
from which we deduce that 1 s ¼ ð1 2r þ r2 Þ ða 1Þ x þ ð1 aÞr : Then the conditions (5) and (6) are satisfied and the orthogonality of the Laguerre polynomials follows. The linear functional L for which we have this orthogonality may be deduced from (16) as detailed below: aþ1 1 Dx ðf Þð0Þ ¼ 1 þ ðf Þð0Þ hL; f i ¼ AðrÞ 1 Dx 1 X aþk a1 Dkx f ð0Þ ðf Þð0Þ ¼ ¼ ð1 Dx Þ k k¼0 Z 1 X 1 Dkx f ð0Þ 1 k a x x x e dx ¼ Cða þ 1Þ k¼0 k! 0 ! Z 1 X 1 1 xk k ¼ Dx f ð0Þ xa ex dx Cða þ 1Þ 0 k! k¼0 Z 1 1 ¼ f ðxÞxa ex dx ðRðaÞ > 1Þ: ð25Þ Cða þ 1Þ 0 Remark. This linear functional L in (25) may also be obtained if we take, as our starting point, another generating function for the Laguerre polynomials (cf., e.g., [12, p. 171, Problem 22 (i)]):
H.M. Srivastava, Y. Ben Cheikh / Appl. Math. Comput. 141 (2003) 415–425
et 0 F 1
423
1 X tn ; xt ¼ LnðaÞ ðxÞ a þ 1; ða þ 1Þn n¼0
ðt 2 CÞ;
where ðkÞn denotes the Pochhammer symbol (or ð1Þn ¼ n! for n 2 N0 ) defined by Cðk þ nÞ 1 ¼ ðkÞn :¼ kðk þ 1Þ ðk þ n 1Þ CðkÞ 1 n!LðaÞ n In fact, according to Lemma 2.4, is ða þ 1Þn n¼0 action of r1 ¼ Dx ðxDx þ aÞ and
ð26Þ
the shifted factorial, since ðn ¼ 0; k 6¼ 0Þ ðn 2 N; k 2 CÞ: quasi-monomial under the 1
s1 ¼ 1 xDx ðxDx þ aÞ D1 x :
1 n!LðaÞ n It follows then that the polynomial set is a r1 -Appell polynomial ða þ 1Þn n¼0 set of the transfer power series A1 ðtÞ ¼ et and that the corresponding linear functional L takes the form: hL; f i ¼ expðr1 Þðf Þð0Þ ¼ expðDx ðxDx þ aÞÞðf Þð0Þ k 1 Dx ðxDx þ aÞ ðf Þð0Þ X ðf 2 PÞ: ¼ k! k¼0
ð27Þ
In particular, for f ðxÞ ¼ xn (n 2 N0 ), we have Cða þ 1 þ nÞ hL; xn i ¼ ða þ 1Þn ¼ Cða þ 1Þ Z 1 1 ¼ xn xa ex dx ðRðaÞ > 1; n 2 N0 Þ: Cða þ 1Þ 0
ð28Þ
3.4. Meixner polynomials The Meixner polynomials Mn ðx; b; cÞ of the first kind are generated by (cf., e.g., [12, p. 449, Problem 20 (ii) with m ¼ 0]) x X 1 c1 t tn b ð1 tÞ ðc 6¼ 0; 1; jtj < 1Þ: 1þ ¼ Mn ðx; b; cÞ c 1t n! n¼0 ð29Þ The generating function (29) is of the type (13) in Lemma 2.5 with x ¼ 1;
b
AðtÞ ¼ ð1 tÞ ;
and
CðtÞ ¼
c1 t ; c 1t
424
H.M. Srivastava, Y. Ben Cheikh / Appl. Math. Comput. 141 (2003) 415–425
from which it is easily seen that t : C ðtÞ ¼ c1 þt c 1 It follows then that the polynomial set fMn ðx; b; cÞgn¼0 is a r-Appell polynomial set of the transfer power series b
AðtÞ ¼ ð1 tÞ ; where r ¼ C ðDÞ. The corresponding raising operator s is given by b 1 þ ðc 1Þx ; s¼ 1r ð1 rÞðc rÞ which may be rewritten as follows: 1 s ¼ c ðc þ 1Þr þ r2 bc c 1 þ ðc 1Þx þ ð2 bÞr ; since we have x
1 1 1 1 ¼ x 1r 1r 1r c1
x
1 1 1 1 ¼ xþ : cr cr cr c1
and
Then the conditions (5) and (6) are satisfied and the orthogonality of the Meixner polynomials follows. The linear functional L for which we have this orthogonality is given by !b 1 D b hL; f i ¼ ðf Þð0Þ ¼ ð1 rÞ ðf Þð0Þ ¼ 1 c1 ðf Þð0Þ AðrÞ þD c 1 b X c ðbÞk c k k D ¼ 1þ ðf Þð0Þ ¼ D f ð0Þ c1 c1 k! k¼0 ! 1 k X k ðbÞk c k X kþj ð 1Þ ¼ f ðjÞ ; c1 k! j k¼0 j¼0 that is, ðbÞkþj c kþj f ðjÞ k!j! c 1 k¼0 j¼0 1 1 X X ðbÞj c j ðb þ jÞk c k ¼ f ðjÞ c1 c1 j! k! j¼0 k¼0
hL; f i ¼
1 X 1 X
¼ ð1 cÞ
ð1Þ
b
j
1 X ðbÞj j c f ðjÞ: j! j¼0
ð30Þ
H.M. Srivastava, Y. Ben Cheikh / Appl. Math. Comput. 141 (2003) 415–425
425
Acknowledgement The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.
References [1] Y. Ben Cheikh, Some results on quasi-monomiality, Appl. Math. Comput. 141 (2003) 63–76. [2] Y. Ben Cheikh, On obtaining dual sequences via quasi-monomiality, Georgian Math. J. 9 (2002) 413–422. [3] R.P. Boas Jr., R.C. Buck, Polynomial Expansions of Analytic Functions (second printing, with corrections), Springer-Verlag, Berlin, 1964. [4] T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. [5] C. Cesarano, Monomiality principle and Legendre polynomials, in: G. Dattoli, H.M. Srivastava, C. Cesarano (Eds.), Advanced Special Functions and Integration Methods (Melfi, 2000), Aracne Editrice, Rome, 2001, pp. 147–164. [6] G. Dattoli, Hermite–Bessel and Laguerre–Bessel functions: a by-product of the monomiality principle, in: D. Cocolicchio, G. Dattoli, H.M. Srivastava (Eds.), Advanced Special Functions and Applications (Melfi, 1999), Aracne Editrice, Rome, 2000, pp. 83–95. [7] G. Dattoli, H.M. Srivastava, C. Cesarano, The Laguerre and Legendre polynomials from an operational point of view, Appl. Math. Comput. 124 (2001) 117–127. [8] G. Dattoli, A. Torre, G. Mazzacurati, Quasi-monomials and isospectral problems, Nuovo Cimento B 112 (1997) 133–138. [9] M.X. He, P.E. Ricci, Differential equations of Appell polynomials via the factorization method, J. Comput. Appl. Math. 139 (2002) 231–237. [10] Y. Smirnov, A. Turbiner, Hidden SL2-algebra of finite difference equations, Modern Phys. Lett. A 10 (1995) 1795–1801. [11] H.M. Srivastava, Some characterizations of Appell and q-Appell polynomials, Ann. Mat. Pura Appl. (Ser. 4) 130 (1982) 321–329. [12] H.M. Srivastava, H.L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, 1984.
Orthogonality of some polynomial sets via quasi-monomiality H.M. Srivastava
a,*
, Y. Ben Cheikh
b
a
b
Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3P4, Canada D epartement de Math ematiques, Facult e des Sciences, Universit e du Centre, TN-5019 Monastir, Tunisia
Abstract The lowering operator r and the raising operator s, associated with a polynomial set fPn g1 n¼0 , are two operators which are independent of n and satisfy the relationships rðPn ÞðxÞ ¼ nPn1 ðxÞ
and sðPn ÞðxÞ ¼ Pnþ1 ðxÞ ðn 2 N0 Þ:
In this paper, we use these operators to study the orthogonality of some polynomial sets. More precisely, we state sufficient conditions, in terms of the r and s operators, to ensure the orthogonality. We also express explicitly, by means of the r operator, the linear functional for which the orthogonality holds true. We obtain some well-known results as particular cases of the results presented in this paper. Ó 2003 Elsevier Science Inc. All rights reserved. Keywords: Orthogonal polynomials; Lowering operators; Raising operators; Linear functionals; Favard theorem; Boas–Buck polynomials; Appell polynomials; Generating functions; Charlier polynomials; Hermite polynomials; Laguerre polynomials; Meixner polynomials
1. Introduction Let P be the vector space of polynomials with coefficients in C. A polynomial sequence fPn g1 n¼0 in P is called a polynomial set if and only if *
Corresponding author. E-mail addresses: [email protected], [email protected] (Y. Ben Cheikh).
[email protected]
(H.M.
0096-3003/03/$ - see front matter Ó 2003 Elsevier Science Inc. All rights reserved. doi:10.1016/S0096-3003(02)00961-X
Srivastava),
416
H.M. Srivastava, Y. Ben Cheikh / Appl. Math. Comput. 141 (2003) 415–425
degðPn Þ ¼ n
ðn 2 N0 :¼ f0; 1; 2; . . .gÞ:
For an integer j, we denote by KðjÞ the space of operators acting on analytic functions that augment (resp. reduce) the degree of every polynomial by ex actly j if j 2 N0 (resp. j 2 Z 0 ), Z0 being the set of nonpositive integers. This tacitly includes the fact that, if r 2 Kð1Þ , then rð1Þ ¼ 0. In an earlier paper [1], it was shown that every polynomial set fPn g1 n¼0 is quasi-monomial. This means that there exists a lowering operator r and a raising operator s, independent of n, such that rðPn ÞðxÞ ¼ nPn1 ðxÞ
and
sðPn ÞðxÞ ¼ Pnþ1 ðxÞ ðn 2 N0 Þ;
ð1Þ
where r and s are in Kð1Þ and Kð1Þ , respectively. Most of the properties of polynomial sets can be deduced by using operational rules with r and s operators. In many works (see, for instance, [5–10]), new and known results related to Hermite polynomials, Laguerre polynomials, Legendre polynomials, and Appell polynomials were derived by using the monomiality principle. The main properties considered in these recent works are related to generating functions, differential equations, and Rodrigues formulas. In this paper, we study the orthogonality of some polynomial sets by using the corresponding lowering and raising operators. In Section 2 of this paper, we recall some useful definitions and results. Then, in Section 3, we present our main result (Theorem 3.1) from which we deduce well-known properties related to the orthogonality of some classical orthogonal polynomials.
2. Basic definitions and results Let P0 be the dual of the vector space P of polynomials with coefficients in C. We denote by hu; f i the effect of the linear functional u 2 P0 on the polynomial f 2 P. Let fPn g1 n¼0 be a polynomial set in P. The corresponding monic 1 b polynomial sequence f P n g is given by n¼0
Pn ¼ kn Pbn
ðn 2 N0 Þ;
where kn is the normalization coefficient and its dual sequence f Pen g1 n¼0 is defined by h Pen ; Pbm i ¼ dn;m
ðn; m 2 N0 Þ;
dn;m being the Kronecker delta. 1 The polynomial sequence fPn gn¼0 is called an orthogonal polynomial set with respect to the functional L if it satisfies the following condition: hL; Pm Pn i ¼ cn dn;m
ðm; n 2 N0 Þ:
ð2Þ
H.M. Srivastava, Y. Ben Cheikh / Appl. Math. Comput. 141 (2003) 415–425
417
The so-called Favard theorem asserts that the orthogonality condition (2) may 1 be deduced from the fact that the sequence f Pbn gn¼0 satisfies a second-order (three-term) recurrence relation [4]: Pbnþ2 ðxÞ ¼ ðx bnþ1 Þ Pbnþ1 ðxÞ cnþ1 Pbn ðxÞ ðn 2 N0 Þ ð3Þ Pb0 ðxÞ ¼ 1 and Pb1 ðxÞ ¼ x b0 and the regularity conditions: ð4Þ cnþ1 6¼ 0 ðn 2 N0 Þ: Sufficient conditions, in terms of lowering and raising operators, to ensure the orthogonality of a polynomial set are given by 1
Lemma 2.1. Let fPn gn¼0 be a polynomial set and let r and s be its lowering and raising operators. If 1 s ¼ a1 þ ða2 þ a3 xÞr þ a4 r2 ða5 þ a6 x þ a7 rÞ; ð5Þ where a1 ; . . . ; a7 are complex parameters constrained by a1 6¼ 0;
ða3 ; a6 Þ 6¼ ð0; 0Þ;
and
ða4 ; a7 Þ 6¼ ð0; 0Þ;
ð6Þ
then the conditions (3) and (4) are satisfied. 1
A polynomial set fBn gn¼0 is called the sequence of basic polynomials for r 2 Kð1Þ if (i) B0 ðxÞ ¼ 1, (ii) Bn ð0Þ ¼ 0 whenever n 2 N :¼ N0 n f0g, (iii) rðBn ÞðxÞ ¼ nBn1 ðxÞ ðn 2 N0 Þ. It is known that every r 2 Kð1Þ has a sequence of basic polynomials, and we have 1
Lemma 2.2 (cf. [2], Lemma 2.2). Let fPn gn¼0 be a polynomial set and let r be its 1 lowering operator. Also let fBn gn¼0 be the sequence of basic polynomials for r. Then there exists a unique power series 1 X uðtÞ ¼ ak tk ða0 6¼ 0Þ k¼0
such that uðrÞðPn ÞðxÞ ¼ Bn ðxÞ
ðn 2 N0 Þ:
We say that fPn g1 n¼0 is a r-Appell polynomial set of the transfer power series AðtÞ, where 1 : AðtÞ ¼ uðtÞ
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According to Lemma 2.2 and another known result [1, Theorem 2.1], every polynomial set can be viewed as a r-Appell polynomial set of the transfer power series AðtÞ for suitable r and AðtÞ, which may be deduced from the following result: 1
Lemma 2.3. A r-polynomial set fPn gn¼0 of the transfer power series AðtÞ is generated by Gðx; tÞ ¼ AðtÞG0 ðx; tÞ ¼
1 X n¼0
Pn ðxÞ
tn ; n!
where G0 ðx; tÞ is a solution of the system: rG0 ðx; tÞ ¼ tG0 ðx; tÞ G0 ðx; 0Þ ¼ 1; and conversely. An explicit expression of the dual sequence of a r-polynomial set fPn g1 n¼0 of the transfer power series AðtÞ was given in [2] by 1 rn rn e ðf ÞðxÞ f ð0Þ ðn 2 N0 ; f 2 PÞ: ¼ ð7Þ hPn; f i ¼ n! AðrÞ n!AðrÞ x¼0 In our present investigation, we need also the following two lemmas (see also [3]). 1
Lemma 2.4 (cf. [1], Corollary 3.3). Let P :¼ fPn gn¼0 be a Boas–Buck polynomial set generated by 1
X tn Gðx; tÞ ¼ AðtÞB xCðtÞ ¼ Pn ðxÞ ; n! n¼0
ð8Þ
where AðtÞ ¼ BðtÞ ¼
1 X k¼0 1 X
ak tk ða0 6¼ 0Þ;
CðtÞ ¼
1 X
ck tkþ1 ðc0 6¼ 0Þ;
and
k¼0
bk tk ðbk 6¼ 0; k 2 N0 Þ:
ð9Þ
k¼0
Also let q 2 Kð1Þ be given by qð1Þ ¼ 0
and
qðxn Þ ¼
bn1 n1 x bn
ðn 2 NÞ
ð10Þ
H.M. Srivastava, Y. Ben Cheikh / Appl. Math. Comput. 141 (2003) 415–425
419
and suppose that q1 ðxn Þ ¼
bnþ1 nþ1 x bn
ðn 2 N0 Þ: 1
Then the polynomial set fPn gn¼0 is quasi-monomial under the action of A0 ðrÞ d 0 1 0 þ xDx C ðrÞq r ¼ C ðqÞ and s ¼ A ðtÞ ¼ Dt AðtÞ; Dz :¼ ; AðrÞ dz ð11Þ where C is the inverse of C, that is,
C CðtÞ ¼ C C ðtÞ ¼ t
with
C ðtÞ ¼
1 X
cn tnþ1
ðc0 6¼ 0Þ:
ck tkþ1
ðc0 6¼ 0Þ
n¼0
Lemma 2.5 (cf. [1], Corollary 3.4). Let AðtÞ ¼
1 X
ak t k
ða0 6¼ 0Þ
and
CðtÞ ¼
k¼0
1 X
ð12Þ
k¼0
be two analytic functions. 1 Then the polynomial set Q :¼ fQn gn¼0 defined the generating function: 1 x=x X tn AðtÞ 1þ xCðtÞ ¼ Qn ðx; xÞ n! n¼0
ð13Þ
is quasi-monomial under the action of r ¼ C ðDx Þ and
s¼
A0 ðrÞ þx AðrÞ
C 0 ðrÞ A0 ðrÞ þ xC 0 ðrÞTx ; ¼ 1 þ xDx AðrÞ
ð14Þ
where Tj is the translation operator and Dx ¼
Tx 1 : x
ð15Þ
3. Orthogonality of a class of polynomial sets Our approach for studying the orthogonality of a polynomial set fPn g1 n¼0 consists of two steps. Firstly, in order to assert the orthogonality of the considered polynomial set, we use the Favard theorem via Lemma 2.1. We then
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take L ¼ Pe0 as the linear functional, for which we have the orthogonality, and we express it by means of (7). We thus obtain our main result given by 1
Theorem 3.1. Let fPn gn¼0 be a r-polynomial set of the transfer power series AðtÞ and let s be its raising operator. If s is given by (5) under the constraints (6), then 1 fPn gn¼0 is an orthogonal polynomial set with respect to the linear functional L given by hL; f i ¼
1 ðf Þð0Þ ðf 2 PÞ: AðrÞ
ð16Þ
We now apply our main result (Theorem 3.1) to some well-known orthogonal polynomial sets. 3.1. Charlier polynomials The monic Charlier polynomials CðaÞ n ðxÞ are generated by ð1 þ tÞx expðatÞ ¼
1 X
CnðaÞ ðxÞ
n¼0
tn n!
ða 6¼ 0; jtj < 1Þ:
ð17Þ
1
It follows then that the polynomial set fCðaÞ n gn¼0 is a D-Appell polynomial set of the transfer power series AðtÞ ¼ eat , where, for convenience, D :¼ D1 . According to Lemma 2.5, fCnðaÞ g1 n¼0 is quasi-monomial under the action of r¼D
s ¼ a þ xT1 :
and
The raising operator s may be rewritten as follows: 1
s ¼ ð1 þ DÞ ð1 a þ x aDÞ; since T1 ¼ 1 þ D
and
T1 x xT1 ¼ T1 :
Then the conditions (5) and (6) are satisfied and the orthogonality of the Charlier polynomials follows. The linear functional L for which we have this orthogonality can be deduced from (16), and we thus obtain hL; f i ¼ expðaDÞðf Þð0Þ ¼
1 X ak k D ðf Þð0Þ k! k¼0
which, in view of the well-known relationship: k X k Dk f ð0Þ ¼ f ðjÞ; ð1Þkþj j j¼0 assumes the form:
ðf 2 PÞ;
ð18Þ
H.M. Srivastava, Y. Ben Cheikh / Appl. Math. Comput. 141 (2003) 415–425
hL; f i ¼
1 X ak k! k¼0
k X
ð 1Þkþj
j¼0
k f ðjÞ j
! ¼
421
1 X 1 X akþj f ðjÞ ð1Þk k!j! k¼0 j¼0
1 X aj ¼ ea f ðjÞ: j! j¼0
ð19Þ
3.2. Hermite polynomials The classical Hermite polynomials Hn ðxÞ are generated by 1 X tn ðt 2 CÞ: Hn ðxÞ expð2xt t2 Þ ¼ n! n¼0
ð20Þ
1
It follows then that the polynomial set fHn gn¼0 is an Appell polynomial set (cf., e.g., [11]) of the transfer power series AðtÞ ¼ expðt2 Þ: According to Lemma 2.4, fHn g1 n¼0 is quasi-monomial under the action of 1 ð21Þ r ¼ Dx and s ¼ 2x Dx ¼ 2x 2r: 2 The expression for the raising operator s in (21) asserts the orthogonality of the Hermite polynomials. The linear functional L for which we have this orthogonality may be deduced from (16) as follows: 1 X 1 D2k x ðf Þð0Þ hL; f i ¼ expðr2 Þðf Þð0Þ ¼ exp D2x ðf Þð0Þ ¼ ðf 2 PÞ: 4 4k k! k¼0 ð22Þ n
In particular, for f ðxÞ ¼ x (n 2 N0 ), we have hL; xn i 8 <0 Z 1 n! 1 nþ1 2 ¼
p ffiffiffi p ffiffiffi ¼ xn expðx2 Þ dx C ¼ : n n 2 p p 0 2 2 ! Z 1 1 p ffiffiffi ¼ xn expðx2 Þ dx: p 1
if n is odd if n is even
ð23Þ 3.3. Laguerre polynomials The Laguerre polynomials LðaÞ n ðxÞ are generated by 1 xt X a1 n exp LðaÞ ðjtj < 1Þ; ð1 tÞ ¼ n ðxÞt 1t n¼0
ð24Þ
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H.M. Srivastava, Y. Ben Cheikh / Appl. Math. Comput. 141 (2003) 415–425
which is of the type (8) in Lemma 2.4 with AðtÞ ¼ ð1 tÞa1
CðtÞ ¼ C ðtÞ ¼
and
t : 1t
1
It follows then that the polynomial set fn!LnðaÞ gn¼0 is quasi-monomial under the action of r¼
Dx 1 Dx
s¼
and
aþ1 x 1r
1 ð1 rÞ
2
:
Using the fact that 1 Dx ¼
1 1r
and
Dx x xDx ¼ 1;
we find that 1 ð1 rÞ
2
xx
1 2
ð1 rÞ
¼
2 ; 1r
from which we deduce that 1 s ¼ ð1 2r þ r2 Þ ða 1Þ x þ ð1 aÞr : Then the conditions (5) and (6) are satisfied and the orthogonality of the Laguerre polynomials follows. The linear functional L for which we have this orthogonality may be deduced from (16) as detailed below: aþ1 1 Dx ðf Þð0Þ ¼ 1 þ ðf Þð0Þ hL; f i ¼ AðrÞ 1 Dx 1 X aþk a1 Dkx f ð0Þ ðf Þð0Þ ¼ ¼ ð1 Dx Þ k k¼0 Z 1 X 1 Dkx f ð0Þ 1 k a x x x e dx ¼ Cða þ 1Þ k¼0 k! 0 ! Z 1 X 1 1 xk k ¼ Dx f ð0Þ xa ex dx Cða þ 1Þ 0 k! k¼0 Z 1 1 ¼ f ðxÞxa ex dx ðRðaÞ > 1Þ: ð25Þ Cða þ 1Þ 0 Remark. This linear functional L in (25) may also be obtained if we take, as our starting point, another generating function for the Laguerre polynomials (cf., e.g., [12, p. 171, Problem 22 (i)]):
H.M. Srivastava, Y. Ben Cheikh / Appl. Math. Comput. 141 (2003) 415–425
et 0 F 1
423
1 X tn ; xt ¼ LnðaÞ ðxÞ a þ 1; ða þ 1Þn n¼0
ðt 2 CÞ;
where ðkÞn denotes the Pochhammer symbol (or ð1Þn ¼ n! for n 2 N0 ) defined by Cðk þ nÞ 1 ¼ ðkÞn :¼ kðk þ 1Þ ðk þ n 1Þ CðkÞ 1 n!LðaÞ n In fact, according to Lemma 2.4, is ða þ 1Þn n¼0 action of r1 ¼ Dx ðxDx þ aÞ and
ð26Þ
the shifted factorial, since ðn ¼ 0; k 6¼ 0Þ ðn 2 N; k 2 CÞ: quasi-monomial under the 1
s1 ¼ 1 xDx ðxDx þ aÞ D1 x :
1 n!LðaÞ n It follows then that the polynomial set is a r1 -Appell polynomial ða þ 1Þn n¼0 set of the transfer power series A1 ðtÞ ¼ et and that the corresponding linear functional L takes the form: hL; f i ¼ expðr1 Þðf Þð0Þ ¼ expðDx ðxDx þ aÞÞðf Þð0Þ k 1 Dx ðxDx þ aÞ ðf Þð0Þ X ðf 2 PÞ: ¼ k! k¼0
ð27Þ
In particular, for f ðxÞ ¼ xn (n 2 N0 ), we have Cða þ 1 þ nÞ hL; xn i ¼ ða þ 1Þn ¼ Cða þ 1Þ Z 1 1 ¼ xn xa ex dx ðRðaÞ > 1; n 2 N0 Þ: Cða þ 1Þ 0
ð28Þ
3.4. Meixner polynomials The Meixner polynomials Mn ðx; b; cÞ of the first kind are generated by (cf., e.g., [12, p. 449, Problem 20 (ii) with m ¼ 0]) x X 1 c1 t tn b ð1 tÞ ðc 6¼ 0; 1; jtj < 1Þ: 1þ ¼ Mn ðx; b; cÞ c 1t n! n¼0 ð29Þ The generating function (29) is of the type (13) in Lemma 2.5 with x ¼ 1;
b
AðtÞ ¼ ð1 tÞ ;
and
CðtÞ ¼
c1 t ; c 1t
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from which it is easily seen that t : C ðtÞ ¼ c1 þt c 1 It follows then that the polynomial set fMn ðx; b; cÞgn¼0 is a r-Appell polynomial set of the transfer power series b
AðtÞ ¼ ð1 tÞ ; where r ¼ C ðDÞ. The corresponding raising operator s is given by b 1 þ ðc 1Þx ; s¼ 1r ð1 rÞðc rÞ which may be rewritten as follows: 1 s ¼ c ðc þ 1Þr þ r2 bc c 1 þ ðc 1Þx þ ð2 bÞr ; since we have x
1 1 1 1 ¼ x 1r 1r 1r c1
x
1 1 1 1 ¼ xþ : cr cr cr c1
and
Then the conditions (5) and (6) are satisfied and the orthogonality of the Meixner polynomials follows. The linear functional L for which we have this orthogonality is given by !b 1 D b hL; f i ¼ ðf Þð0Þ ¼ ð1 rÞ ðf Þð0Þ ¼ 1 c1 ðf Þð0Þ AðrÞ þD c 1 b X c ðbÞk c k k D ¼ 1þ ðf Þð0Þ ¼ D f ð0Þ c1 c1 k! k¼0 ! 1 k X k ðbÞk c k X kþj ð 1Þ ¼ f ðjÞ ; c1 k! j k¼0 j¼0 that is, ðbÞkþj c kþj f ðjÞ k!j! c 1 k¼0 j¼0 1 1 X X ðbÞj c j ðb þ jÞk c k ¼ f ðjÞ c1 c1 j! k! j¼0 k¼0
hL; f i ¼
1 X 1 X
¼ ð1 cÞ
ð1Þ
b
j
1 X ðbÞj j c f ðjÞ: j! j¼0
ð30Þ
H.M. Srivastava, Y. Ben Cheikh / Appl. Math. Comput. 141 (2003) 415–425
425
Acknowledgement The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.
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