A CHARACTERIZATION OF ULTRASPHERICAL POLYNOMIALS 1. THE QUESTION

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1 A CHARACTERIZATION OF ULTRASPHERICAL POLYNOMIALS MICHAEL ANSHELEVICH ABSTRACT. We show that the only orthogonal polynomials with a generating function of the form F xz αz are the ultraspherical, Hermite, and Chebyshev polynomials of the first kind. The generating function for the Chebyshev case is non-standard, although it is easily derived from the usual one.. THE QUESTION Hermite polynomials H n x are one of the most important families of orthogonal polynomials in mathematics, appearing in probability theory, mathematical physics, differential equations, combinatorics, etc. One of the simplest ways to construct them is through their generating function, n! H nxz n = exp xz z /. On the other hand, Chebyshev polynomials of the second kind U n x are another important family of orthogonal polynomials appearing in numerical analysis, for example, with a generating function U n xz n = xz + z. Note that both of these functions have the form F xz αz, with F z = e z, respectively, F z = z. Question. What are all the orthogonal polynomials with generating functions of the form F xz αz for some number α and function or, more precisely, formal power series F? A reader interested in further context and background for this question may want to start by reading Section 5.. THE METHOD The following is the first fundamental theorem about orthogonal polynomials. Theorem Favard s theorem. Let {P 0 x, P x, P x,...} be a monic orthogonal polynomial family. That is, Date: August, Mathematics Subject Classification. Primary 33C45; Secondary 4C05, 46L54. This work was supported in part by the NSF grant DMS

2 MICHAEL ANSHELEVICH Each P n is a polynomial of degree exactly n polynomial family. Each P n x = x n + lower order terms monic. For some non-decreasing function M with lim x Mx = 0, lim x Mx =, and infinitely many points of increase equivalently, for some probability measure supported on infinitely many points the Stieltjes integral for all n k orthogonal. P n xp k x dmx = 0 Then there exist real numbers {β 0, β, β,...} and positive real numbers {ω, ω, ω 3,...} such that the polynomials satisfy a three-term recursion relation xp n = P n+ + β n P n + ω n P n to make the formula work for n = 0, take P = 0. Actually, Favard s theorem also asserts that the converse to the statement above is true: if we have a monic polynomial family satisfying such a recursion, these polynomials are automatically orthogonal for some Mx, and the case of M with only finitely many points of increase can also be included. See [Chi78, Chapter ] or [Ism05, Chapter ] for the proof. We only need the easy direction stated above, which we now prove. Proof. Every polynomial P of degree k is a linear combination of {P i : 0 i k}. Therefore P is orthogonal to all P n with k < n. Since we can expand xp n x = x n+ + lower order terms, xp n x = P n+ x + c n,n P n x + c n,n P n x c n, P x + c n,0 P 0 x for some coefficients c n,n,..., c n,0. On the other hand, for k < n orthogonality implies c n,k P k xp k x dmx = xp n x P k x dmx = P n x xp k x dmx = 0 since deg xp k < n. Since M has infinitely many points of increase, P k cannot be zero at all of those points, and as a result Therefore c n,k = 0 for k < n, so in fact P k x dmx > 0. xp n x = P n+ x + c n,n P n x + c n,n P n x

3 A CHARACTERIZATION OF ULTRASPHERICAL POLYNOMIALS 3 only. Denoting β n = c n,n and ω n = c n,n, we get the formula. It remains to note that c n,n P n xp n x dmx = P n x xp n x P n+ x c n,n P n x dmx = = = P n x xp n x dmx xp n x P n x dmx P n xp n x dmx. Since both P n x dmx and P nx dmx are positive, so is ω n = c n,n. 3. EXAMPLES Good references for polynomial families are Wikipedia and [KS98]. Beware that in the following examples, we use for monic polynomials the notation which usually appears for other normalizations, so our formulas may differ from the references by a re-scaling. Example. The orthogonality relation for the Hermite polynomials is H n xh k x e x / dx = 0. π Thus they are orthogonal with respect to the normal Gaussian distribution. They satisfy a recursion xh n x = H n+ x + nh n x. Example. The orthogonality relation for the Charlier polynomials is C n ic k i e = 0, i! so that which is sometimes also written as i=0 [x] Mx = e i!, dmx = e i=0 i=0 i! δ ix. Thus the Charlier polynomials are orthogonal with respect to the Poisson distribution. They satisfy a recursion xc n x = C n+ x + n + C n x + n C n x. Example 3. The Legendre polynomials are the family of orthogonal polynomials a student is most likely to encounter in an undergraduate course. In a linear algebra course, one sees them in the applications of the Gram-Schmidt formula, since their orthogonality relation is simply P n xp k x dx = 0.

4 4 MICHAEL ANSHELEVICH In a differential equations course, one sees them as a solution of the Legendre equation. This equation, in turn, arises after the separation of variables in the heat equation or Laplace s equation in three dimensions, or on a sphere. The recursion satisfied by the Legendre polynomials is xp n x = P n+ x + n 4n P n x Example 4. The ultraspherical also called Gegenbauer polynomials C n λ are orthogonal with respect to Mx = x λ with integration restricted to [, ] for λ >, λ 0. Note that the Legendre polynomials are a particular case corresponding to λ =. Just as the Legendre polynomials are related to the sphere, the other ultraspherical families for half-integer λ are related to higher-dimensional spheres. They satisfy a recursion 3 xc n λ x = C n+x λ nn + λ + 4n + λ n + λ Cλ n x, see Section.8. of [KS98]. Another important special case are the Chebyshev polynomials of the second kind U n, which correspond to λ = and are orthogonal with respect to x. For λ = 0, the coefficient ω as written in formula 3 is undefined. The polynomials orthogonal with respect to x are the Chebyshev polynomials of the first kind T n, for which the recursion 3 holds for n with λ = 0, but xt x = T x + T 0x. Remark. Suppose {P n } form a monic orthogonal polynomial family, with orthogonality given by a function M and the recursion 4 xp n = P n+ + β n P n + ω n P n. Then for r 0, the polynomials Q n x = r n P n x/r are also monic, orthogonal with respect to Nx = Mx/r, and satisfy Indeed, from equation 4, xq n = Q n+ + rβ n Q n + r ω n Q n. r n+ x/rp n x/r = r n+ P n+ x/r + β n r n+ P n x/r + ω n r n+ P n x/r, which implies the recursion for {Q n }. 4. THE RESULT Theorem. Let α > 0 and F z = c nz n be a formal power series with c 0 =, c = c 0. Define the polynomials {P n : n 0} via 5 F xz αz = c n P n xz n if c n = 0, P n is undefined. These polynomials form an orthogonal polynomial family which is automatically monic if and only if

5 A CHARACTERIZATION OF ULTRASPHERICAL POLYNOMIALS 5 {P n } are re-scaled ultraspherical polynomials, P n x = C λ n b 4α x, for λ >, λ 0, and b > 0. In this case F z = + c λb bz λ for c 0. The choice c = λb gives simply F z = bz λ. {P n } are re-scaled Chebyshev polynomials of the first kind, b P n x = T n 4α x, for b > 0. In this case F z = + c b ln bz for c 0. The choice c = b gives simply F z = + ln bz. {P n } are re-scaled Hermite polynomials, for a > 0. In this case a P n x = H n α x F z = + c a eaz for c 0. The choice c = a gives simply F z = e az. Remark. If equation 5 holds, so that then clearly also + + c n xz αz n = F xz αz = + Cc n xz αz n = FC xz αz = + c n P n xz n, Cc n P n xz n for any C 0 and F C z = + CF z. This is the source of the free parameter c in the theorem.

6 6 MICHAEL ANSHELEVICH Proof of the Theorem. Using the binomial formula, expand F xz αz = = c n xz αz n c n x n z n = + c n nxz n αz +... c n x n αn c n x n z n +... If for some n, c n = 0 while c n 0, then comparing the coefficients of z n in the preceding equation and expansion 5, we see that the coefficient is non-zero on the left and zero on the right. So all c n 0, and we may denote d n = cn c n. Then the same coefficient comparison gives P 0 x =, and for n P n x = x n αn d n x n +... Using this equation for both n and n +, we then get xp n = x n+ αn d n x n +... = P n+ αn d n x n + αnd n+x n +... If we want the polynomials to be orthogonal, by Favard s theorem they have to satisfy a three-term recursion relation xp n = P n+ + β n P n + ω n P n note that {P n } are clearly monic. We see that β n = 0, and ω n = α nd n+ n d n for n. Now expanding further in the binomial formula, c n xz αz n = c n x n z n Thus and for n 4, c n nxz n αz + n= nn c n xz n αz +... = + c xz + c x αc z + c 3 x 3 αc x z 3 + c n x n αn c n x n n n 3 + α c n x n 4 z n +... n=4 P 0 x =, P x = x, P x = x αd, P 3 x = x 3 αd 3 x P n x = x n αn d n x n + α n n 3 d n d n x n

7 Therefore for n 4, xp n P n+ ω n P n = A CHARACTERIZATION OF ULTRASPHERICAL POLYNOMIALS 7 α n n 3 α n d n nd n+ d n d n α a formula which also holds for n = 3. For this to be zero we need n 3 d n d n n d n+d n or n 3 d n+ n n x n , αn d n n d n nd n+ n d n n d n+ nd n = 0. Thus for n 3, n + d n+ = nd n n d n. The general solution of this recursion is nd n = a + bn for n. Since all d n 0, a, b cannot both be zero. Therefore for n and If a 0, b 0, then 6 If a = 0, b 0, then c n = F z = + c 7 F z = + c a + bn c n = n F z = + cz + c n i=0 n i= n= a + ib c = n! n i= a/b i bz n an! n i= a + ib z n. n! d n = 0, a + ib c n! = + c bz a/b = + c a a bz. a/b b n n zn = c b ln bz = + c b ln, bz d n+d n which can also be obtained from the preceding formula by using L Hôpital s rule. Finally, if a 0, b = 0, then a n 8 F z = + c z n = + c n! a eaz. Moreover, n b + a ω n = αn n b + anb + a. Since for orthogonality, we need ω n 0, clearly b 0. If b = 0, then ω n = α a n > 0

8 8 MICHAEL ANSHELEVICH as long as a > 0. The polynomials with this recursion re-scaled Hermite polynomials. We recall [KS98, Section.3] that the generating function for standard monic Hermite polynomials is n! H nxz n = exp xz z /, which is of the form 5 with F z = e z, and the generating function 8 is obtained from it by a re-scaling and a shift from Remark. If b > 0, a 0, we denote λ = a/b and get Since we have λ >. Since ω n = α b nn + λ n + λ n + λ. ω = α b + λ, ω = α + λ b + λ + λ, we have moreover λ >. It is now easy to see that this condition suffices for the positivity of all ω n ; indeed, the corresponding polynomials are re-scaled ultraspherical polynomials. We recall [KS98, Section.8.] that the generating function for standard monic ultraspherical polynomials is n n i=0 n! λ i C λ n xz n = xz + z λ, which is of the form 5 with F z = z λ, and the generating function 6 is obtained from it by a re-scaling and a shift from Remark. Finally, if b > 0, λ = a = 0, then ω n = α b for n, but ω = α. These are precisely recursion coefficients for the re-scaled Chebyshev b polynomials of the fist kind. The standard generating function [KS98, Section.8.] for monic Chebyshev polynomials of the first kind is n T n xz n xz = xz + z, so it is not of the form 5. However, n T n xz n = z xz xz + z = Term-by-term integration with respect to z gives n C + n T nxz n = ln xz + z = ln x z xz + z. xz z / with C = 0, which is of the form 5 with F z = ln z. The generating function 7 is obtained from it by a re-scaling and a shift from Remark..

9 A CHARACTERIZATION OF ULTRASPHERICAL POLYNOMIALS 9 5. THE HISTORY The question of characterizing various classes of orthogonal polynomials has a long and distinguished history, see [AS90] for an excellent survey up to 990. The study of general polynomial families goes back to Paul Appell in 880 [App80], who looked at polynomials with generating functions of the form n! P nxz n = Az expxz for some function Az. These are now called Appell polynomials. Later, they were generalized to Sheffer families with generating functions 9 n! P nxz n = Az expxuz for some functions Az, Uz. The prototypical orthogonal polynomials characterization result is Meixner s 934 description of all orthogonal polynomials with the Sheffer-type generating functions [Mei34]. On the other hand, among Appell polynomials, only Hermite polynomials are orthogonal. See Figure. Sheffer Orthogonal Appell Meixner H FIGURE. The relationship between the class of orthogonal polynomials and Sheffer, Appell and Meixner classes. H stands for Hermite. Example 5. One way to state Meixner s result is that orthogonal polynomials with generating functions 9 satisfy a three-term recursion xp n x = P n+ x + na + β 0 P n x + nn b + P n x, for some a, b, β 0 up to re-scaling. The function M a,b,β0 for which these polynomials are orthogonal can be written down explicitly, but for different values of the parameters these functions look quite different. For example, for a = b = β 0 = 0, we get the Hermite polynomials, with a continuous orthogonality relation. On the other hand, for a = β 0 =, b = 0, we get the Charlier polynomials, with a discrete orthogonality relation. Other polynomials in the Meixner class carry the names of Laguerre, Krawtchouk, Meixner, and Pollaczek, and are orthogonal with respect to gamma, binomial, negative binomial, and Meixner distributions. Besides nice generating functions, the Meixner class has many other characterizations and applications, see [DKSC08] for an excellent but advanced survey. Perhaps for this reason, many generalizations of this class have been attempted. The most popular of these are probably the q-deformed

10 0 MICHAEL ANSHELEVICH families. One approach there are several extends the Sheffer class by looking at the generating functions of the form Az qu q k z z k=0 after appropriate normalization, one gets the Sheffer form for q. In this case the analog of the Meixner class are the Al-Salam and Chihara polynomials [ASC87]. For the study of two different types of q-appell polynomials, see [AS67, AS95]. A different generalization of the Sheffer class are generating functions of the general Boas-Buck [BB64] type: c n P n xz n = AzF xuz for F z = c nz n with c 0 =. The usual case corresponds to F z = e z. In the Boas-Buck setting, the problem of describing all orthogonal polynomials is wide open. The Appell-type class with Uz = z in this case consists of the Brenke polynomials, and at least in that case all the orthogonal polynomials are known [Chi68]. Now note that in the Sheffer/Meixner case in equation 9, corresponding to F z = e z, the generating function has an alternative form n! P nxz n = Az expxuz = expxuz + log Az. So another interesting class to look at are all or just orthogonal polynomials with generating functions c n P n xz n = F xuz Rz, which again gives the Sheffer/Meixner families for F z = e z. The case F z = appears in Free Probability [NS06], see Section 3 of [Ans03] for the author s z description of the free Meixner class, which is in a precise bijection with the Meixner class except for the binomial case [BB06]. Here again, one can write the generating function in two ways: Az xuz =. x Uz More generally, Boas and Buck proved the following result. Az Az Az Theorem. [BB56] The only functions F with F 0 = such that 0 AzF xuz = F xuz Rz are F z = e z and F z = z λ for some λ. So as an alternative to the Boas-Buck formulation, we are interested in orthogonal polynomials with generating functions of the form F xuz Rz, or at least in the Appell-type subclass F xz Rz. For general R, even this seems to be a hard question. However, the orthogonal Appell polynomials are only the Hermite polynomials, with the exponential generating function exp xz z /.

11 A CHARACTERIZATION OF ULTRASPHERICAL POLYNOMIALS On the other hand, the orthogonal free Appell polynomials are only the Chebyshev polynomials of the second kind, with the ordinary generating function xz z. Moreover, Rz = αz appears naturally in combinatorial proofs of the usual, free, and other central-limit-type theorems see for example Lecture 8 of [NS06]. Thus it is reasonable to consider F xz αz first, which leads to Question. Conversely, the answer to that question indicates that interesting generating functions and also, potentially, interesting non-commutative probability theories arise precisely for F covered by the Boas-Buck theorem above, plus in the exceptional case F z = + log not covered by that theorem. On the other hand, see [ASV86, Dem09] for z some negative results. Acknowledgments. I am grateful to Harold Boas for a very careful reading of the paper, and for many valuable comments, including Remark. REFERENCES [AS67] Waleed A. Al-Salam, q-appell polynomials, Ann. Mat. Pura Appl , MR #6670 [AS90], Characterization theorems for orthogonal polynomials, Orthogonal polynomials Columbus, OH, 989, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 94, Kluwer Acad. Publ., Dordrecht, 990, pp. 4. MR0086 9g:40 [AS95], A characterization of the Rogers q-hermite polynomials, Internat. J. Math. Math. Sci , no. 4, MR i:3303 [ASC87] W. A. Al-Salam and T. S. Chihara, q-pollaczek polynomials and a conjecture of Andrews and Askey, SIAM J. Math. Anal , no., 8 4. MR a:3309 [ASV86] W. A. Al-Salam and A. Verma, Some sets of orthogonal polynomials, Rev. Técn. Fac. Ingr. Univ. Zulia 9 986, no., MR895 88e:33009 [Ans03] Michael Anshelevich, Free martingale polynomials, J. Funct. Anal , no., 8 6. MR f:46079 [App80] M. P. Appell, Sur une classe de polynômes, Ann. Sci. Ecole Norm. Sup , [BB56] Ralph P. Boas, Jr. and R. Creighton Buck, Polynomials defined by generating relations, Amer. Math. Monthly , MR ,300e [BB64], Polynomial expansions of analytic functions, Second printing, corrected. Ergebnisse der Mathematik und ihrer Grenzgebiete, N.F., Bd. 9, Academic Press Inc., Publishers, New York, 964. MR #8 [BB06] Marek Bożejko and Włodzimierz Bryc, On a class of free Lévy laws related to a regression problem, J. Funct. Anal , no., MR79 007a:4607 [Chi68] T. S. Chihara, Orthogonal polynomials with Brenke type generating functions, Duke Math. J , MR #307 [Chi78], An introduction to orthogonal polynomials, Gordon and Breach Science Publishers, New York, [Dem09] 978, Mathematics and its Applications, Vol. 3. MR #979 Nizar Demni, Ultraspherical type generating functions for orthogonal polynomials, Probab. Math. Statistics , no., [DKSC08] Persi Diaconis, Kshitij Khare, and Laurent Saloff-Coste, Gibbs sampling, exponential families and orthogonal polynomials, Statist. Sci , no., 5 00, With comments and a rejoinder by the authors. MR [Ism05] Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 005, With two chapters by Walter Van Assche, With a foreword by Richard A. Askey. MR f:3300

12 MICHAEL ANSHELEVICH [KS98] [Mei34] [NS06] Roelof Koekoek and René F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Tech. Report 98-7, Delft University of Technology, Department of Technical Mathematics and Informatics, 998, koekoek/askey/. J. Meixner, Orthogonale polynomsysteme mit einer besonderen gestalt der erzeugenden funktion, J. London Math. Soc , 6 3. Alexandru Nica and Roland Speicher, Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, vol. 335, Cambridge University Press, Cambridge, 006. MR k:4698 DEPARTMENT OF MATHEMATICS, TEXAS A&M UNIVERSITY, COLLEGE STATION, TX address: manshel@math.tamu.edu