A CHARACTERIZATION OF ULTRASPHERICAL POLYNOMIALS
MICHAEL ANSHELEVICH
ABSTRACT. We show that the only orthogonal polynomials with a generating function of the form F xz − αz2 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.
1. THEQUESTION
Hermite polynomials Hn(x) are one of the most important families of orthogonal polynomials in mathematics, appearing in probability theory, mathematical physics, differential equations, combi- natorics, etc. One of the simplest ways to construct them is through their generating function, ∞ X 1 H (x)zn = exp xz − z2/2 . n! n n=0
On the other hand, Chebyshev polynomials of the second kind Un(x) are another important fam- ily of orthogonal polynomials (appearing in numerical analysis, for example), with a generating function ∞ X 1 U (x)zn = . n 1 − xz + z2 n=0 Note that both of these functions have the form F (xz − αz2), with F (z) = ez, respectively, F (z) = 1 1−z . Question 1. What are all the orthogonal polynomials with generating functions of the form F xz − αz2 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.
2. THEMETHOD
The following is the first fundamental theorem about orthogonal polynomials.
Theorem (Favard’s theorem). Let {P0(x),P1(x),P2(x),...} be a monic orthogonal polynomial family. That is,
Date: August 1, 2011. 2010 Mathematics Subject Classification. Primary 33C45; Secondary 42C05, 46L54. This work was supported in part by the NSF grant DMS-0900935. 1 2 MICHAEL ANSHELEVICH
• Each Pn is a polynomial of degree exactly n (polynomial family). • Each n Pn(x) = x + lower order terms (monic). • For some non-decreasing function M with limx→−∞ M(x) = 0, limx→∞ M(x) = 1, and infinitely many points of increase (equivalently, for some probability measure supported on infinitely many points) the Stieltjes integral Z ∞ Pn(x)Pk(x) dM(x) = 0 −∞ for all n 6= k (orthogonal).
Then there exist real numbers {β0, β1, β2,...} and positive real numbers {ω1, ω2, ω3,...} such that the polynomials satisfy a three-term recursion relation
xPn = Pn+1 + βnPn + ωnPn−1
(to make the formula work for n = 0, take P−1 = 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 M(x), and the case of M with only finitely many points of increase can also be included. See [Chi78, Chapter 1] or [Ism05, Chapter 2] 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 {Pi : 0 ≤ i ≤ k}. Therefore P is orthogonal to all Pn with k < n. Since
n+1 xPn(x) = x + lower order terms, we can expand
xPn(x) = Pn+1(x) + cn,nPn(x) + cn,n−1Pn−1(x) + ... + cn,1P1(x) + cn,0P0(x) for some coefficients cn,n, . . . , cn,0. On the other hand, for k < n − 1 orthogonality implies Z ∞ Z ∞ Z ∞ cn,k Pk(x)Pk(x) dM(x) = xPn(x) Pk(x) dM(x) = Pn(x) xPk(x) dM(x) = 0 −∞ −∞ −∞ since deg (xPk) < n. Since M has infinitely many points of increase, Pk cannot be zero at all of those points, and as a result Z ∞ 2 Pk (x) dM(x) > 0. −∞
Therefore cn,k = 0 for k < n − 1, so in fact
xPn(x) = Pn+1(x) + cn,nPn(x) + cn,n−1Pn−1(x) A CHARACTERIZATION OF ULTRASPHERICAL POLYNOMIALS 3 only. Denoting βn = cn,n and ωn = cn,n−1, we get the formula. It remains to note that Z ∞ Z ∞ cn,n−1 Pn−1(x)Pn−1(x) dM(x) = Pn−1(x) xPn(x) − Pn+1(x) − cn,nPn(x) dM(x) −∞ −∞ Z ∞ = Pn−1(x) xPn(x) dM(x) −∞ Z ∞ = xPn−1(x) Pn(x) dM(x) −∞ Z ∞ = Pn(x)Pn(x) dM(x). −∞ R ∞ 2 R ∞ 2 Since both −∞ Pn−1(x) dM(x) and −∞ Pn (x) dM(x) are positive, so is ωn = cn,n−1.
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 normaliza- tions, so our formulas may differ from the references by a re-scaling. Example 1. The orthogonality relation for the Hermite polynomials is Z ∞ 1 −x2/2 (1) Hn(x)Hk(x) √ e dx = 0. −∞ 2π Thus they are orthogonal with respect to the normal (Gaussian) distribution. They satisfy a recursion
xHn(x) = Hn+1(x) + nHn−1(x). Example 2. The orthogonality relation for the Charlier polynomials is ∞ X 1 (2) C (i)C (i) e−1 = 0, n k i! i=0 so that [x] X 1 M(x) = e−1 , i! i=0 which is sometimes also written as ∞ X 1 dM(x) = e−1 δ (x). i! i i=0 Thus the Charlier polynomials are orthogonal with respect to the Poisson distribution. They satisfy a recursion xCn(x) = Cn+1(x) + (n + 1) Cn(x) + n Cn−1(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 Z 1 Pn(x)Pk(x) dx = 0. −1 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 n2 xP (x) = P (x) + P (x) n n+1 4n2 − 1 n−1 (λ) Example 4. The ultraspherical (also called Gegenbauer) polynomials Cn are orthogonal with respect to 2 λ− 1 M(x) = (1 − x ) 2 1 (with integration restricted to [−1, 1]) for λ > − 2 , λ 6= 0. Note that the Legendre polynomials are a 1 particular case corresponding to λ = 2 . 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 n(n + 2λ − 1) (3) xC(λ)(x) = C(λ) (x) + C(λ) (x), n n+1 4(n + λ − 1)(n + λ) n−1 see Section 1.8.1 of [KS98]. Another important special case are the Chebyshev√ polynomials of the 2 second kind Un, which correspond to λ = 1 and are orthogonal with respect to 1 − x .
For λ = 0, the coefficient ω1 as written in formula (3) is undefined. The polynomials orthogonal with respect to √ 1 are the Chebyshev polynomials of the first kind T , for which the recursion 1−x2 n (3) holds for n ≥ 2 (with λ = 0), but 1 xT (x) = T (x) + T (x). 1 2 2 0
Remark 1. Suppose {Pn} form a monic orthogonal polynomial family, with orthogonality given by a function M and the recursion
(4) xPn = Pn+1 + βnPn + ωnPn−1. Then for r 6= 0, the polynomials n Qn(x) = r Pn(x/r) are also monic, orthogonal with respect to N(x) = M(x/r), and satisfy 2 xQn = Qn+1 + rβnQn + r ωnQn−1. Indeed, from equation (4), n+1 n+1 n+1 n+1 r (x/r)Pn(x/r) = r Pn+1(x/r) + βnr Pn(x/r) + ωnr Pn−1(x/r), which implies the recursion for {Qn}.
4. THE RESULT
P∞ n Theorem 1. Let α > 0 and F (z) = n=0 cnz be a formal power series with c0 = 1, c1 = c 6= 0. Define the polynomials {Pn : n ≥ 0} via ∞ 2 X n (5) F xz − αz = cnPn(x)z n=0
(if cn = 0, Pn is undefined). These polynomials form an orthogonal polynomial family (which is automatically monic) if and only if A CHARACTERIZATION OF ULTRASPHERICAL POLYNOMIALS 5
•{Pn} are re-scaled ultraspherical polynomials, r ! b P (x) = C(λ) x , n n 4α
1 for λ > − 2 , λ 6= 0, and b > 0. In this case
c 1 F (z) = 1 + − 1 λb (1 − bz)λ
1 for c 6= 0. The choice c = λb gives simply F (z) = (1−bz)λ . •{Pn} are re-scaled Chebyshev polynomials of the first kind, r ! b P (x) = T x , n n 4α
for b > 0. In this case c 1 F (z) = 1 + ln b 1 − bz