PROBABILISTIC ASPECTS OF AL-SALAM–CHIHARA POLYNOMIALS W×ODZIMIERZ BRYC, WOJCIECH MATYSIAK, AND PAWE×J. SZAB×OWSKI Abstract. We solve the connection coe¢ cient problem between the Al-Salam– Chihara polynomials and the q-Hermite polynomials, and we use the resulting identity to answer a question from probability theory. We also derive the distribution of some Al-Salam–Chihara polynomials, and compute determinants of related Hankel matrices.
1. Introduction and main identity The aim of the paper is to point out the connection of Al-Salam–Chihara polynomials with a regression problem in probability, and to use it to give a new simple derivation of their density. Our approach exploits identity (1.8) below, which connects the Al-Salam–Chihara polynomials to the continuous q-Hermite polynomials. This connection is more direct and elementary but less general than the technique of attachment exploited in [BI96, Section 2]. We also compute determinants of Hankel matrices with entries which are linear combinations of the q-Hermite polynomials. The Al-Salam–Chihara polynomials were introduced in [ASC76]. We are interested in the renormalized Al-Salam–Chihara polynomials fpn (xjq; a; b)g, which are de…ned by the following three term recurrence relation (1.1)
pn+1 (x) = (x
aq n ) pn (x)
with the usual initial conditions p standard notation
1 1 (x)
bq n
1
[n]q pn
1
(x) (n
0);
= 0, p0 (x)(x) = 1. Here, we use the
[n]q = 1 + q + + qn [n]q ! = [1]q [2]q : : : [n]q ; [n]q ! n = k q [n k]q ![k]q !
1
;
with the usual conventions [0]q = 0; [0]q ! = 1. For jqj < 1, their generating function f (t; xjq; a; b) =
1 X tn pn (xjq; a; b) [n]q ! n=0
Date : April 11, 2003 . 1991 Mathematics Subject Classi…cation. Primary: 33D45 Secondary: 05A30, 15A15, 42C05. Key words and phrases. q-Hermite polynomials, matrix of moments, orthogonal polynomials, determinants, polynomial regression. 1
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W ×ODZIM IERZ BRYC, WOJCIECH M ATYSIAK, AND PAW E×J. SZAB×OW SKI
is given by (1.2)
f (t; xjq; a; b) =
1 Y 1 1
q)atq k + (1 q)bt2 q 2k ; q)xtq k + (1 q)t2 q 2k
(1 (1
k=0
compare [AI84, (3.6) and (3.10)]. The corresponding (renormalized) continuous q-Hermite polynomials Hn (xjq) = pn (xjq; 0; 0) satisfy the three term recurrence relation (1.3)
Hn+1 (x) = xHn (x)
[n]q Hn 1 (x): P1 tn Hn (xjq) is For jqj < 1 their generating function (t; xjq) = n=0 [n] q! (1.4)
(t; xjq) =
1 Y
1
(1
q)xtq k + (1
q)t2 q 2k
1
:
k=0
Of course, these are well known special cases of (1.1) and (1.2), see [ISV87, (2.11) and (2.12)] which we state here for further reference. We will also use polynomials fBn (xjq)g de…ned by the three term recurrence relation (1.5)
Bn+1 (x) =
q n xBn (x) + q n
1
[n]q Bn
1
(x) (n
0)
with the usual initial conditions B 1 = 0; B0 = 1. These polynomials are related to the q-Hermite polynomials by ( p in q n(n 2)=2 Hn (i q xj 1q ) if q > 0 p : (1.6) Bn (xjq) = 1 n(n 1)=2 n(n 2)=2 ( 1) jqj Hn ( jqj xj q ) if q < 0 Indeed, both sides of (1.6) are equal at n = 1; 0 and satisfy (1.5) for n P1 tn Bn (xjq) is given by Their generating function (t; xjq) = n=0 [n] q!
(1.7)
(t; xjq) =
1 Y
1
(1
q)xtq k + (1
0.
q)t2 q 2k :
k=0
To see this, we compute the q-derivative for t 6= 0; q 6= 1 and use (1.5) to get 1 (t; xjq) (qt; xjq) X tn = Bn+1 (xjq) = (1 q)t [n]q ! n=0 x Thus
1 1 X X (qt)n (qt)n Bn (xjq) + t Bn (xjq) = ( x + t) (qt; xjq): [n]q ! [n]q ! n=0 n=0
(t; xjq) = (tq; xjq) 1 n Y n (tq ; xjq) 1 (1
(1
q)xt + (1
q)xtq k + (1
q)t2 =
q)t2 q 2k ;
k=0
and (1.7) follows since (q n t; xjq) ! (0; xjq) = 1 as n ! 1 when jqj < 1. We now point out the mutual relationship between the Al-Salam–Chihara polynomials fpn (xjq; a; b)g and the polynomials fHn (xjq)gn 0 and fBn (xjq)gn 0 . Theorem 1. For all a; b; q 2 C, b 6= 0, and n 1 we have n X a n (1.8) pn (xjq; a; b2 ) = bn k Bn k ( jq) Hk (xjq) k q b k=1
a bk Hk ( jq) b
PROBABILISTIC ASPECTS OF AL-SALAM –CHIHARA POLYNOM IALS
3
Proof. From recurrence relations (1.1), (1.3), and (1.5), it is clear that pn (xjq; a; b), Hn (xjq), and Bn (xjq) are given by polynomial expressions in variable q. The qn binomial coe¢ cient is also a polynomial in q. Therefore, we see that identity k q (1.8) is equivalent to a polynomial identity in variable q 2 C. Hence it is enough to prove that (1.8) holds true for all jqj < 1. When jqj < 1, inspecting (1.2), (1.4), and (1.7) we notice that f (t; xjq; a; b2 ) = (tb; a=bjq) (t; xjq);
(1.9) and (1.10)
(t; xjq) (t; xjq) = 1:
Therefore, f (t; xjq; a; b2 ) = 1 + (bt; a=bjq) ( (t; xjq)
(bt; a=bjq)) ;
which is valid for all small enough jtj. Comparing the coe¢ cients at tn for n and taking into account that Hk (xjq) bk Hk ( ab jq) = 0 for k = 0, we get (1.8).
1
Remark 1. One could split (1.8) into the following two separate identities, which are implied by (1.9) and (1.10) respectively: (1.11)
(1.12)
8n
0 : pn (xjq; a; b2 ) =
n X
k=0
8n
1:
n X n Bn k q
n k
bn
k
Bn
k(
q
k (x)Hk
a jq)Hk (xjq); b
(xjq) = 0:
k=0
Formula (1.11) is a renormalized inverse of formula [IRS99, (4.7)], who express the q-Hermite polynomials as linear combinations of Al-Salam–Chihara polynomials. Formula (1.12) resembles [Car56, (2.28)], who considers q-Hermite polynomials of Pn n the form hn (xjq) = k=0 xk , paired with bn (xjq) = hn (xj1=q). k q 2. Probabilistic aspects Quadratic regression questions in the paper [Bry01] lead to the problem of determining all probability distributions which are de…ned indirectly by the relationships Z (2.1) Hn (xjq) (dx) = n Hn (yjq) n = 1; 2; : : : ; where y; ; q 2 R are …xed parameters, and fHn gn 0 is the family of the q-Hermite polynomials. Our next result shows that this problem can be solved using the Al-Salam– Chihara polynomials. Theorem 2. If = (dxj ; y) satis…es (2.1), then its orthogonal polynomials are Al-Salam–Chihara polynomials fpn (xjq; a; b)g with a = y, b = 2 .
4
W ×ODZIM IERZ BRYC, WOJCIECH M ATYSIAK, AND PAW E×J. SZAB×OW SKI
Proof. Recall that Hn (xjq) = pn (xjq; 0; 0). Thus if = 0 then (2.1) implies that R pn (xjq; a; b) (dx) = 0 for all n = 1; 2; : : : . Suppose now that 6= 0. Combining (1.8) with (2.1) we get Z Z n X n n k k pn (xjq; a; b) (dx) = Bn k (yjq) Hk (xjq) Hk (yjq) (dx) = 0 k q k=0
for R all n = 1; 2; : : : . Since fpn g satisfy a three step recurrence, this implies pk (x)pn (x) (dx) = 0 for all 0 k < n. Next we answer an unresolved case from [Bry01].
Corollary 1. If ; q; y 2 R are such that q > 1 and probabilistic solution .
6= 0 then (2.1) has no
Proof. Suppose that is positive and solves (2.1). Then its monic orthogonal polynomials satisfy the three term recurrence relation (1.1) (2.2)
pn+1 (x) = (x
yq n )pn (x)
2 n 1
(1
q
)[n]q pn
1 (x):
For a positive measure, the coe¢ cients at the third term in the recurrence must be 2 n 1 non-negative, 1 q 0 for all n, which is false. From Theorem 2 it follows that if the solution of (2.1) exists, then it is given by the distribution of the Al-Salam-Chihara polynomials. The distribution of the Al-Salam-Chihara polynomials is derived in [AI84, Chapter 3]. However, in [Bry01, Proposition 8.1] we found the solution of (2.1) which relies solely on the facts about the q-Hermite polynomials. We repeat the latter argument here, and then use it to re-derive the distribution of the corresponding Al-Salam–Chihara polynomials. Corollary 2. If ; q; y 2 R are such that j j the probabilistic solution of (2.1) is given by with the density on x2 < 4=(1 q) given by p 1 2 k Y (1 q ) 1 1 q p 2 q 2k )2 2 (1 (1 q) 2 4 (1 q)x k=0
< 1,jqj < 1, and y 2 (1 q) < 4, then the absolutely continuous measure q k+1 (1 + q k )2 (1 q)x2 q k q k (1 + 2 q 2k )xy + (1 q) 2 (x2 + y 2 )q 2k
Proof. The distribution of the q-Hermite polynomials Hn (xjq) is supported on x2 < 4=(1 q) with the density p 1 1 Y Y 1 q p fH (x) = (1 + q k )2 (1 q)x2 q k (1 q k+1 ); 2 4 (1 q)x2 k=0 k=0
see [ISV87, (2.15)]. Moreover, since jHn (x)j 4=(1 q), the series (2.3)
gH (x; y; ) =
1 X
n=0
Cq (n + 1)(1
q)
n=2
when x2 ; y 2
n
[n]q !
Hn (x)Hn (y)
converges uniformly and de…nes the Poisson-Mehler kernel which is given by (2.4) 1 2 k Y (1 q ) gH (x; y; ) = ; 2 q 2k )2 (1 (1 q) q k (1 + 2 q 2k )xy + (1 q) 2 (x2 + y 2 )q 2k k=0
PROBABILISTIC ASPECTS OF AL-SALAM –CHIHARA POLYNOM IALS
5
this is the renormalized version of the well known result, see eg. [IS88, (2.2)] who p consider the q-Hermite polynomials given by f(1 q)n=2 Hn (2x= 1 qjq)g instead of our fHn (xjq)g. R Since (1.3) implies that Hn2 (xjq)fH (x)dx = [n]q !, it follows from (2.3) that Z 2=p1 q Hn (xjq)gH (x; y; )fH (x) dx = n Hn (y): p 2= 1 q
Corollary 3. If q; a; b 2 R are such that jqj < 1, 0 < b < 1, and a2 (1 q) < 4b, then the distribution of the Al-Salam-Chihara polynomials fpn (xjq; a; b)g is absolutely continuous with the density on x2 < 4=(1 q) given by p 1 Y (1 bq k ) 1 q k+1 (1 + q k )2 (1 q)x2 q k 1 q p : (1 bq 2k )2 (1 q)aq k (1 + bq 2k )x + (1 q)(bx2 + a2 )q 2k 2 4 (1 q)x2 k=0
Proof. By Theorem 2, the distribution of polynomials pn solves (2.1) with p b; y = a= . Thus the formula follows from Corollary 2.
=
3. Determinants of Hankel matrices In this section we are interested in calculating the determinants of the Hankel matrices Mn = [mi+j ]i;j=0;:::n 1 ; R i where mi = x (dx) are the moments of certain (perhaps signed) measure . It is well known that for positive measures we must have det Mn 0, and that these determinants can be read out from the three term recurrence for the corresponding monic orthogonal polynomials. R Consider …rst the moments mk (y) = xk (dx) of the (perhaps signed) measure = y; , which solves (2.1). Then mk (y) are polynomials of degree k in variable y, and can be written as follows. Let an;2i , i bn=2c be the coe¢ cients in the expansion of monomial xn into the q-Hermite polynomials, xn =
bn=2c
X
an;2i Hn
2i
(xjq) ; n
0:
i=0
Then mn (y) =
Z
xn d (x) =
bn=2c
X
n 2k
an;2k Hn
2k
(yjq) :
k=0
Let Sn be the Hankel matrix of moments mk (y), 2 m0 (y) m1 (y) : : : 6 m1 (y) m2 (y) 6 Sn (yjq; ) = 6 .. .. 4 . . mn
1 (y)
mn
: : : m2n
1 (y)
.. . 2 (y)
3
7 7 7: 5
It is well known that det Sn is the product of the coe¢ cients at the third term of (2.2), which implies the following. Qn 2 i 1 Corollary 4. det Sn+1 = det Sn = [n]q ! i=1 1 q :
6
W ×ODZIM IERZ BRYC, WOJCIECH M ATYSIAK, AND PAW E×J. SZAB×OW SKI
Our second Hankel matrix has an even simpler form. As indicated in [IS97], [IS02] the q-Hermite polynomials can be viewed as moments of a signed measure, R Hn (xjq) = un (dujx; q). It turns out that measure (dujx; q) cannot be positive even for a single value of x. To see this, consider the following n n matrices 3 2 H0 (xjq) H1 (xjq) H2 (xjq) : : : Hn 1 (xjq) 6 H1 (xjq) H2 (xjq) Hn (xjq) 7 7 6 6 7 .. 6 . Hn+1 (xjq) 7 Mn (xjq) = 6 H2 (xjq) 7 6 7 .. .. .. 5 4 . . . Hn 1 (xjq) Hn (xjq) : : : H2n 2 (xjq)
The following q-generalization of [Kra99, (3.55)] shows that the determinants det Mn (xjq) are free of the variable x, and take negative values. Theorem 3.
det Mn+1 = ( 1)n q n(n det Mn
1)=2
[n]q !
Proof. Using (1.3), we row-reduce the …rst column of the matrix. Namely, from the second row of Mn+1 , we subtract the …rst one multiplied by x. Similarly, for i 3, we subtract x times row i 1 and add the i 2-th row multiplied by [i 1]q . Taking (1.3) into account, det Mn+1 (xjq) becomes 2 H0 H1 H2 ::: Hn 6 0 ([0] [1])H ([0] [2])H ([0] [n])Hn 1 0 1 6 6 0 ([1] [2])H ) ([1] [3])H ([1] [n + 1])Hn 1 2 det 6 6 .. . . .. .. 4 . 0
([n
1]
[n])Hn
([n
1
Now, we use the fact that for m [n]q
Thus det Mn+1 (xjq) becomes 2 H0 H1 6 0 [1]H 0 6 6 0 q[1]H 1) det 6 6 .. 4 . 0
q
n 1
[1]Hn
1]
[n + 1])Hn
: : : ([n
1]
[2n
1])H2n
n we have
[m]q = q m [n
m]q :
H2 [2]H1 q[2]H2
:::
..
1
q
n 1
[2]Hn
. :::
Hn 1 [n]Hn 1 q[n]Hn .. . qn
1
[n]H2n
3 7 7 7 7 7 5
Expanding det Mn+1 with respect to the …rst column, and factoring out the common factors q i 1 from the i-th row and [j]q from the j-th column of the resulting matrix, we get n Pn 1 Y n i i=1 [j] det Mn det Mn+1 = ( 1) q j=1
=
n n(n 1)=2
( 1) q
[n]q ! det Mn :
Formula stated in Corollary 4 was originally discovered through symbolic computations and motivated this paper. We were unable to …nd a direct algebraic proof along the lines of the proof of Theorem 3, and our search for the explanation of
3 7 7 7 7 7 5
PROBABILISTIC ASPECTS OF AL-SALAM –CHIHARA POLYNOM IALS
7
why det Sn (y) does not depend on y lead us to Al-Salam–Chihara polynomials and identity (1.8). The the fact that Hankel determinants formed of certain linear combinations of the q-Hermite polynomials do not depend on the argument of these polynomials as exposed in Theorem 3 and Corollary 4 is striking and unexpected to us. A natural question arises whether other linear combinations have this property. Acknowledgement Part of the research of WB was conducted while visiting Warsaw University of Technology. The authors thank Jacek Weso÷ owski for several helpful discussions. References [AI84]
Richard Askey and Mourad Ismail. Recurrence relations, continued fractions, and orthogonal polynomials. Mem. Amer. Math. Soc., 49(300):iv+108, 1984. [ASC76] W. A. Al-Salam and T. S. Chihara. Convolutions of orthonormal polynomials. SIAM J. Math. Anal., 7(1):16–28, 1976. [BI96] Christian Berg and Mourad E. H. Ismail. q-Hermite polynomials and classical orthogonal polynomials. Canad. J. Math., 48(1):43–63, 1996. [Bry01] W÷odzimierz Bryc. Stationary random …elds with linear regressions. Annals of Probability, 29:504–519, 2001. [Car56] L. Carlitz. Some polynomials related to theta functions. Ann. Mat. Pura Appl. (4), 41:359–373, 1956. [IRS99] Mourad E. H. Ismail, Mizan Rahman, and Dennis Stanton. Quadratic q-exponentials and connection coe¢ cient problems. Proc. Amer. Math. Soc., 127(10):2931–2941, 1999. [IS88] Mourad E. H. Ismail and Dennis Stanton. On the Askey-Wilson and Rogers polynomials. Canad. J. Math., 40(5):1025–1045, 1988. [IS97] Mourad E. H. Ismail and Dennis Stanton. Classical orthogonal polynomials as moments. Can. J. Math., 49(3):520–542, 1997. [IS02] Mourad E. H. Ismail and Dennis Stanton. q-integral and moment representations for q-orthogonal polynomials. Canad. J. Math., 54(4):709–735, 2002. [ISV87] M. E. H. Ismail, D. Stanton, and G. Viennot. The combinatorics of q-Hermite polynomials and the Askey-Wilson integral. Europ. J. Combinatorics, 8:379–392, 1987. [Kra99] Christian Krattenhaler. Advanced determinant calculus. Sém. Lothar. Combin., 42:Art. B42q, 67 pp. (electronic), 1999. E-mail address :
[email protected] E-mail address :
[email protected] E-mail address :
[email protected] Department of Mathematics, University of Cincinnati, P.O. Box 210025, Cincinnati, OH 45221–0025 Faculty of Mathematics and Information Science, Warsaw University of Technology, pl. Politechniki 1, 00-661 Warszawa, Poland Faculty of Mathematics and Information Science, Warsaw University of Technology, pl. Politechniki 1, 00-661 Warszawa, Poland
