ULTRASPHERICAL TYPE GENERATING FUNCTIONS FOR ORTHOGONAL POLYNOMIALS
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1 ULTRASPHERICAL TYPE GENERATING FUNCTIONS FOR ORTHOGONAL POLYNOMIALS arxiv:083666v [mathpr] 8 Jan 009 Abstract We characterize, up to a conjecture, probability distributions of finite all order moments with ultraspherical type generating functions for orthogonal polynomials Motivation: Meixner families There is a one to one correspondance between probability distributions on the real line and polynomials of a one variable satisfying a three-terms recurrence relation subject to some positive conditions ([9]) That is why in most of the cases, if not all, one tries to characterize probability distributions using generating functions for orthogonal polynomials Among the famous generating functions are the ones of exponential type, that is if µ is a probability distribution with a finite exponential moment in a neighborhood of zero e zx µ(dx) <, then () ψ(z, x) : R P n (x)z n exh(z) E(e XH(z) ), where H is analytic around z 0 such that H(0) 0, H (0), X is a random variable in some probability space (Ω, F, P) with law µ P X and (P n ) is the set of orthogonal polynomials with respect to µ Up to translations and dilations, there are six probability distributions which form the so-called Meixner family referring to its first appearance with J Meixner ([4]) It consists of Gaussian, Poisson, Gamma, negative binomial, Meixner and binomial distributions This family appeared many times under differents guises ([6], [3], [], [5], []) Another well known example was first suggested and studied in [] and is given by a Cauchy-Stieltjes type kernel Namely, if µ is a probability distribution of finite all order moments, then () ψ(z, x) : P n (x)z n u(z)[f(z) x] where u and z zf(z) are analytic functions around zero such that u(z) lim lim zf(z) z 0 z z 0 This family, known as the free Meixner family due to its intimate relation to free probability theory, covers six compactly-supported probability measures too We Key words and phrases Generating functions, ultraspherical type, orthogonal polynomials, Jacobi-Szegö parameters
2 refer the reader to [4], [5], [8], [] for more characterizations and more interpretations The natural q-deformation that interpolates the forementioned families for arbitrary q was defined and studied in [3] and is up to affine transformations the so-called Al-Salam and Chihara family of orthogonal polynomials ([]) Their generating functions is given by an infinite product and is somehow similar to the q-exponential function Another characterization of the last family was recently given in [7] After this sketchy overview, we suggest another type of generating functions which may be viewed as a generalization of the free Meixner family It is inspired from the case of Gegenbauer or ultraspherical polynomials for which ([9]) (3) n() n Cn n! (x)zn ( zx + z, x, > 0, ) where () n ( + n ) ( + ) and for complex z such that the RHS makes sense and the series in the LHS converges We adopted here the monic normalization for (C n) n and henceforth all the polynomials are monic so that they satisfy the normalized recurrence relation (4) xp n (x) P n+ (x) + α n P n (x) + ω n P n (x), n 0, P : 0, ω 0 The sequences (α n ), (ω n ) are known as the Jacobi-Szegö parameters and ω n > 0 for all n unless µ is has a finite support ([9]) It is then natural to adress the problem of characterizing probability measures of finite all order moments, say µ, such that (5) ψ (z, x) : () n Pn (x)z n n! u (z)(f (z) x), > 0 for x supp(µ ) and such that u is analytic in a neighborhood of zero cut along the negative real axis, z zf (z) is analytic in a neighborhood of zero and lim zf (z), z 0 u (z) lim z 0 z Moreover, we require that (f (z) x) avoids negative integers for every x R Note that the above limiting conditions ensure that ψ (0, x) for all x supp(µ ) We shall say that ψ is a generating function for orthogonal polynomials of ultraspherical-type referring to ultraspherical polynomials Without loss of generality, we may assume that µ is standard, that is, has a zero mean and a unit variance Equivalently, if (α n ), (ωn ) denote the Jacobi-Szegö parameters of µ, then one has α 0 0, ω Our strategy is based on the following general remark that was stated in [6]: Claim: to a given generating function for orthogonal polynomials (z, x) ψ(z, x) associated with a (standard) probability measure µ satisfying an integrability condition (for instance finite all order moments or finite exponential moment), there corresponds a one-parameter family of probability measures such that the variance is a polynomial of degree in the mean Namely, if ψ may be expanded as a convergent infinite series ψ(z, x) a n P n (x)z n
3 for some fixed sequence (a n ) n, x supp(µ) and z in a suitable complex domain, such that (6) a n P n z n <, P n : [P n (x)] µ(dx), then P z (dx) : ψ(z, x)µ(dx) defines a one parameter family of probability measures such that the first and the second moments of P z are at most linear and at most quadratic polynomials in z Indeed, Cauchy-Schwartz inequality, (6) and Fubini s Theorem allow to integrate ψ termwise and the orthogonality of P n implies that P z is a probability measure for all z Moreover, by the same arguments and the finiteness of both the second and the fourth moments, one may exchange the order of integration in (7) a n (xp n (x))z n µ(dx), a n (x P n (x))z n µ(dx) Now, the orthogonality of P n implies a n P n+ (x)µ(dx)z n 0 a n ω n P n (x)µ(dx)z n a n α n P n (x)µ(dx)z n a 0 α 0 0, a ω a z so that one gets after using (4) a n (xp n (x))z n µ(dx) a z xψ(z, x)µ(dx) For the second moment of ψ(z, ), one uses twice (4) to get (8) a n (x P n (x))z n µ(dx) a ω z + a α z + x ψ(z, x)µ(dx) The family P z is then referred to as a ψ-family of an at most quadratic variance referring to exponential and Cauchy-Stieltjes families ([5], [8]) When ψ is handable enough so that one can perform computations of the first and of the second moments of P z, one recovers two equations that may be used to solve the problem of characterization of probability measures whose generating function for orthogonal polynomials is of ψ-type In the case of the Meixner and the free Meixner families, this was noticed in [6] In the present case, both equations allow to derive a nonlinear one order differential equation for f and our main result is Proposition () Assume that ψ given by (5) makes sense for z in some open complex domain, then f satisfies Q (z)f (z) f (z) Q (z)f (z) + R (z) where Q, R are polynomials of degree while Q is a polynomial of degree Moreover the coefficients of these polynomials depend only on, α, ω 3
4 () The function u is related to f by Once we did, we show that if u (z) u (z) f (z) f (z) z (9) f (z) : E (z) z where E is assumed to be a polynomial, then deg(e ) and this follows from the fact that Q, Q, R are polynomials (terminating series) Next, we investigate under the above assumption the case of symmetric measures We show that there exist two families of probability measures: (Cn) n for > 0 and (Cn ) n for > /, We warn the reader to the fact that, though these two families coincide, their generating functions given by (5) are totally different since a n depends on and is fixed for both families Under the same assumption, there is only one family of non symmetric probability measures corresponding to shifted monic Jacobi polynomials Pn /, 3/ for > /, The discard of the value is needed for the computations since we need to remove factors like, Thus, one deals with this case separately and recovers the free Meixner family for which deg(e ) Problems: we do not know if there exists a solution f for which E is an entire infinite series Besides, we already know that the free Meixner family ( ) covers six families of probability distributions ([4]) and that E is a polynomial, while there are three families for when E is a polynomial Is there any intuitive explanation to this difference? Proof of the Proposition First and second moments First, let us proceed to the compuations of the first and the second moments On the one hand, one gets from (7) and (8) m (z) : xψ (z, x)µ(dx) z, m (z) : x ψ (z, x)µ(dx) ( + ) ω z + α z + On the other hand, the integration of both sides of (5) with respect to µ gives u (z) (f (z) x) µ (dx) R by orthogonality of P n Then, using the elematary operation x (x f(z))+f(z), it follows that m (z) f(z) u,(z) u (z), u,(z) : (f(z) x) µ (dx) With the help of ( )f (z)u (z) (u, ) (z) substituted in the RHS of m (z), one gets: (0) which can be written as R u (z) u (z) f (z) f (z) z () (u (z)[f (z) z)]) ( )u (z)f (z) 4
5 For the second moment, use x x(x f(z)) + xf(z) to get () m (z) zf (z) x u (z) (f (z) x) µ (dx) Note that ( ) x (f (z) x) µ (dx) ( )f (z) R so that () implies that R R x (f (z) x) µ (dx) ( )zu (z)f (z) (3) ( [zf (z) m (z)]u (z) ) ( )zu (z)f (z) A non linear differential equation By the virtue of (), (3) implies that ( [zf (z) m (z)]u (z) ) z(u (z)[f (z) z)]) which gives therefore [zf (z) m (z)]u (z) + [f (z) + zf (z) (m ) (z)]u (z) z[f (z) z]u (z) + z[f (z) )]u (z), [ z m (z)]u (z) [(m ) (z) f (z) z]u (z) If z m (z) 0, one gets after the comparison of the last equality to (0) (m ) (z) f (z) z z m (z) f (z) f (z) z which shows after elemantary computations that f satisfies the following non linear first order differential equation: (4) where Q (z)f (z) f (z) Q (z)f (z) + R (z) [ Q (z) + ω Q (z) ( + )ωz + α, ( + ) R (z) ω z Setting g (z) : f (z) [Q (z)/], (4) transforms to ] z α z, (5) Q (z)g (z) g (z) + Q (z) where Q (z) R (z) 4 [Q (z)] + ω Q (z) [( + )ω ] 4 ω z + α ω z + ( + )ω (α ) 4 Finally, once g is given, one deduces f by adding Q / then use (0) to derive u 5
6 3 One solution Now, we shall look for a solution of the form g (z) : E (z), E (0) z for a second degree polynomial E In fact, since z zg (z) is analytic around zero, one may always assume that g (z) has the above form for an entire function E But if E is a polynomial of degree 3, then all the terms of degree 3 will vanish only by equating both sides of (5) For instance, let and write (5) as E (z) a 0 z 3 + a z + a z + a 3 (6) Q (z)[ze (z) E (z)] E (z) z Q (z) Then by equating terms of degree 6 is this equation, one easily gets a 0 0 so that E has degree For E a polynomial of degree 4, start with equating terms of degree 4 and so on However, this way of thinking fails or rather become cumbersome when E is an entire function Remark (Free Meixner family) Note that Q is a constant polynomial for so that (5) transforms for this parameter value to [( ω )z α z ]g (z) g (z) + (ω ) (α ) /4 But it is well known that if µ belongs to the free Meixner family, than its Jacobi- Szegö parameters are given by ([4]) α n a, a R, n, ω n ( + b), b, n, where we used the fact that µ has a mean zero (α 0 0) and a unit variance (ω ) Moreover, one has ([5]) + az + ( + b)z f (z) g (z) (a/)z + a z z + z It is then an easy exercice to check that g satisfies (5) which reads in this case (7) [bz + az + ]g (z) g (z) + b a /4 3 Symmetric measures: ultraspherical polynomials In the sequel, we shall focus on the case α n 0 for all n This is equivalent to the fact that µ is symmetric, that is the image of µ by the map x x is still µ In this case, one gets by taking α 0 Q (z) [ ( + )ω ]z, Q (z) [( + )ω ] ω 4 z + ( + )ω Writing E (z) a 0 z + a z + a and equating both sides in (5), one gets: a, a 0, 3a 0 [ ( + )ω ] ( + )ω, a 0 + a 0 [ ( + )ω ] [( + )ω ] ω 4 6
7 The third equation gives a 0 ( )( ω ) 6 Hence, it remains to check when the above a 0 satisfies the fourth equation For, this gives a 0 0 and one has a unique solution For and after substituting a 0 in the fourth equation, one can remove the term ( ) and see that ω satisfies ( + )( + )(ω ) + (4 + 6 )ω + ( 4 ) 0 What is quite interesting and even surprising, that though this polynomial looks complicated, its descriminant is equal 9 so that there are two solutions given by ω + + ω + where for the second value, we consider > / in order to avoid finitely-supported probability measures and signed measures As a result, Thus and from (0) Finally a 0 ( + ), a 0 ( + ) f (z) + z + z, f (z) z + z, u (z) u (z) z, u (z) z, > 0,, u (z) u (z) u (z) + (/)z z z( (/)z ) for suitable complex numbers z such that u makes sense z, > /,, (/)z 3 Orthogonal polynomials The first value of ω corresponds to the Gegenbauer polynomials as we shall see and the Jacobi-Szegö parameters are well known ([9]) However in order to fit into our setting, one has to consider the monic Gegenbauer polynomials which are orthogonal with respect to the standard Beta distribution c ( x /[( + )]) / dx, x [± ( + )] for some normalizing constant c Easy computations show that these polynomials are given by ( ) (8) C n (x) : [( + )] n/ Cn x ( + ) and that their Jacobi-Szegö parameters are given by: (9) α n 0, n 0, ω n n( + )(n + ) (n + )(n + ), n In fact, The Jacobi-Szegö parameters of (C n) n are given by ([0]): α n 0, n 0, ω n n(n + ) 4(n + )(n + ), n 7
8 Hence, on the one hand, it follows that ω + + On the other hand, it is easy to see from (3) that the generating function for the polynomials in (8) is given by () n C n! n(x)z n ( ) ( n () )n n Cn x + z n! ( + ) ( zx + ( + )z /) z [ + ( + )z / z For the second value, ψ is written as: ψ (z, x) x] u (z)(f (z) x) (/)z z (z/ + /z x) (/)z (z / + zx) and we claim that P n C n for all n and all > / In fact, different ways lead to this claim For instance, () n n! C n (x)z n + n ( ) n n! ( )z z ( )z z [ (/)z ( zx + z /) C n (x)z n ( ) n n! z zx + z / C n (x)z n+ ] as the reader may easily check Another way uses the non monic Gegenbauer polynomial (often denoted by Cn ) defined by (0) Ĉn : () n n Cn n! However, one first needs to express P n through C n or C n To proceed, use the generating function for ( C n ) n to see that P n (x) B n (x) with P0 and P (x) x, where B n (x) : ( + n(n ) ( + n )( + n ) B n (x) : B n(x) c n, B n (x), n, ) n C n ( + x 8 ) ( ) n C n ( ) x
9 Now elemantary computations show that n() n Pn n! (x) ( ) [Ĉ n n (n + ) Ĉn ( ) ( )] x x Ĉ n ( ) x n() n C n, n! where we used equation (450) p 95 in [9] to derive the second equality 4 non-symmetric probability measures Henceforth, we suppose that α 0 and we will show that there are two families of probability measures correponding to Then, we get the following equations g (z) a 0z + a z + a z a, a α 0, 3a 0 [ ( + )ω ] a ( + )ω (α ), 4 a 0 α a 0 a α ω, a 0 + a 0 [ ( + )ω ] [( + )ω ] ω 4 From the second, third and fourth equations, it follows that a 0 [ (α ) ] + ω 6 4 ω Actually, this gives a constraint on, α, ω: ( (α () ) ) + ( + 3 ) ω Substituting a 0 by ( )ω /(4) and assuming, the fifth equation becomes 6 (ω ) + ω 8 [ ( + )ω ] [ ( + )ω ]ω 4 In the non degenerate case ω 0, ω But is a double root of the polynomial in the denominator so that ω 3 ( + ) ( /), 9
10 which is positive for > / Finally, one deduces from () that [ (α ( + 3) ] ) ( + ) ( /) ( + ) ( /) > 0, It follows that a 0 ( ) ( + ) ( /) ( ) ( + )( ) f (z) a 0z + a z + a + ( + )ω z + α z [( ) + z + ω z + + ] α + [ ] z z ± z + and u (z) u (z) ] [ ( ) ( ) [ z z z ± z + ] The descriminant of the polynomial ( ) z ± z + is easily seen to be: 4( ) > 0 It follows that, when α > 0, the roots are given by z, z Writing [ ][ ( ) ( z ) z + z one gets u (z) u (z) [ ] [ ] z + z + z ], z z + / As a result z u (z) z + / and the generating function is written as ][ ( ) ] ψ (z, x) [z + z x + z Using the generating function for the monic Gegenbauer polynomials C n, one gets [ ( ) ] [ ( ) z x + z z x + 0 () n Kn n! (x)zn ( + )( /) ] + z
11 where Kn is the shifted Gegenbauer polynomial given by [ ] n [ Kn(x) : ( + )( /) C n ( + )( /) As a result, ( x Pn (x) K n (x) + n n + K n (x), n, with P0 In the case α < 0, similar computations yield z u (z) z / and ψ (z, x) [z ] [ ( z x + ) ] ) ] + z One finally gets Pn (x) Sn(x) n n + S n (x), n, with P0, where [ ] n [ Sn (x) : ( + )( /) C n ( + )( /) ( x + ) ] 4 Jacobi-Szegö parameters Recall that the Jacobi-Szegö parameters of ( C n ) n are given by α n 0, n 0, ω n n( + )(n + ) (n + )(n + ), n Then, the ones of the polynomials (K n) n are given by α n, n 0, ω n while the ones of (S n) n are given by n(n + ) (n + )(n + ), n, α n, n 0, ωn n(n + ) (n + )(n + ), n Tedious computations show that the Jacobi-Szegö parameters of (P n ) n are given by if α > 0 and α n ω n n(n + ) (n + )(n + ), n 0, n(n + ) (n + ), n α n(n + ) n (n + )(n + ), n 0, ω n n(n + ) (n + ), n
12 if α < 0 Since (α n ) n, (ω n ) n are uniformly bounded sequences, then µ is compactlysupported ([9]) 4 Orthogonality measures: Jacobi polynomials We will show that P n is expressed through a shifted Jacobi polynomials with parameters depending on To proceed, we need some rescaling since we assumed through our paper that the orthogonality measure has a mean zero and a unit variance Consider the case α > 0 and split α n as α n ( ) + (n + )(n + ) where we assumed that so that α 0 0 It follows that the monic polynomials [ ] n ( ) Rn (x) Pn x + are monic orthogonal polynomials and their Jacobi-Szegö parameters are given by α n (n + )(n + ) n 0, ωn n(n + ) 4 (n + ), n Now, recall that ([0]) the monic Jacobi polynomials Pn α,β are orthogonal with respect to the Beta distribution with density function given by c α,β ( x) α ( + x) β [,] (x), α, β >, for some normalizing constant c α,β, and that their Jacobi-Szegö parameters are given by α n β α 4 [n + (α + β)/][n + (α + β)/ + ] n 0, ωn n(n + α)(n + β)(n + α + β) 4 [n + (α + β )/][n + (α + β + )/][n + (α + β)/], n Substituting α / > 0, β 3/ >, one easily sees that (α+β+)/ α, (α + β )/ β, (α + β)/ and (β α )/4 ( )/ so that [ ] n ( ) Rn (x) Pn x + Pn /, 3/ (x) or equivalently [ ] n ( ) Pn (x) x Pn /, 3/ It follows that Pn are orthogonal with respect to ) / ( x x µ (dx) c ( + for some normalizing constant c and for [ ] + x, ) 3/ dx
13 The case α < 0 is dealt with similarly and one should take α 3/, β / to see that [ ] n ( ) Pn (x) x + Pn 3/, / Both cases are related using Pn α,β (x) ( ) n Pn β,α ( x) ([9]): ( ) ( ) x + ( x) Pn 3/, / ( ) n Pn /, 3/ References [] W Al-Salam, T S Chihara Convolutions of orthogonal polynomials SIAM J Funct Anal 7, 976, 6-8 [] M Anshelevich Free martingale polynomials J Funct Anal 0, 003, 8-6 [3] M Anshelevich Appell polynomials and their relatives Int Math Res Not 65, 004, [4] M Bozejko, W Bryc On a class of free Lévy laws related to a regression problem J Funct Anal 36, no, 006, [5] M Bozejko, N Demni Generating functions of Cauchy-Stieltjes type for orthogonal polynomials To appear in Infinite Dimen Anal Quantum Probab Relat Top [6] M Bozejko, N Demni Topics on Meixner families Submitted to The Proceedings of the -th Workshop on Noncommutative Harmonic Analysis with Applications to Probability Poland, Bedlewo, 008 [7] W Bryc, M E H Ismail Approximation operators, q-exponential and free exponential families Available on arxiv [8] W Bryc, M E H Ismail Cauchy-Stieltjes kernel families Available on arxiv [9] M E H Ismail Classical and Quantum orthogonal polynomials in one variable Cambridge University Press 005 [0] R Koekoek, R F Swarttouw The Askey Scheme of Hypergeometric Orthogonal Polynomials and its q-analogue Available at itstudelftnl/ koekoek/askey [] I Kubo Generating functions of exponential-type for orthogonal polynomials Inf Dimens Anal, Quantum Probab Related Topics 7, no, 004, [] I Kubo, H H Kuo, S Namli The characterization of a class of probability measures by multiplicative renormalization Communications on Stochastic Analysis, no 3 007, [3] R G Laha, E Lukacs On a problem connected with quadratic regression Biometrika 47, 960, [4] J Meixner Orthogonale polynomsysteme mit einer besonderern der erzeugenden funktion J London Math Soc 9, 934, 6-3 [5] C N Morris Natural exponential families with quadratic variance function Ann Stat 0, no 98, [6] I M Sheffer Some properties of polynomial set of type zero Duke Math J 5, 939,
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