THE ORTHOGONALITY OF GEGENBAUER POLYNOMIALS C_{n}^{λ}(x) FOR λ < −n

KATHY DRIVER AND HELMUT PRODINGER

The Gegenbauer (or ultraspherical) polynomials C_{n}^{λ}(z) can be defined by the generating function

(1 − 2zr + r^{2})^{−λ} = ∑_{n=0}^{∞} C_{n}^{λ}(z) r^{n};

see [1]. For λ > −1/2, these polynomials C_{n}^{λ}(z) are orthogonal on the interval (−1, 1) with respect to the weight function (1 − z^{2})^{λ−1/2} and therefore have all of their zeros in that interval. As the parameter λ decreases below −1/2, the zeros of C_{n}^{λ}(z) depart from (−1, 1) and when λ < 1 − n, all the zeros of C_{n}^{λ}(z) lie on the imaginary axis (cf. [2]).

An interesting question is whether there is some underlying orthogonality that gives rise to the location of the zeros of C_{n}^{λ}(z) on the imaginary axis for λ < 1 − n. We show that this is indeed the case for λ < −n by proving that the sequence of polynomials {i^{n}C_{n}^{λ}(ix)}_{n=1}^{∞} is orthogonal on (−∞, ∞) with respect to the weight function (1 + x^{2})^{λ−1/2}. Note that for a given parameter λ, the set of polynomials is finite (1 ≤ n < −λ). This orthogonality property does not appear to hold for −n < λ < 1 − n. Since C_{n}^{λ}(z) is an even/odd function of z according to whether n is even/odd, the sequence of polynomials {i^{n}C_{n}^{λ}(ix)} is real–valued for x real, while the weight function (1+x^{2})^{λ−1/2} is positive for all x. It therefore follows from the classical theory of real orthogonal polynomials that all zeros of C_{n}^{λ}(ix) are real for λ < −n. Thus the zeros of C_{n}^{λ}(z) are pure imaginary for λ < −n.

As an application of the orthogonality of {i^{n}C_{n}^{λ}(ix)} for λ < −n, we derive an expression for

∫_{−∞}^{∞} C_{l}^{λ}(ix) C_{m}^{λ}(ix) C_{n}^{λ}(ix) (1 + x^{2})^{λ−1/2} dx

which is analogous to the result given in [1, p. 321, eq. (6.8.10)] for λ > −1/2.

Theorem 1. For n, m < −λ we have

∫_{−∞}^{∞} i^{n} C_{n}^{λ}(ix) i^{m} C_{m}^{λ}(ix) (1 + x^{2})^{λ−1/2} dx = g_{n}δ_{nm} (1)

with

g_{n} = \frac{(−1)^{n}2^{−2λ−1}[(λ)_{n}]^{2}[Γ(−n − λ)]^{2}}{n! (n + 2λ)_{n} Γ(−2n − 2λ)} . (2)

Date: January 3, 2002.

Proof. The Rodrigues formula for the Gegenbauer polynomials is given by (cf. [1, p. 303, eq. (6.4.14)])

(1 − x^{2})^{λ−1/2}C_{n}^{λ}(x) = \frac{(−2)^{n}(λ)_{n}}{n! (n + 2λ)_{n}} D^{n} [(1 − x^{2})^{n+λ−1/2}] . (3)

Replacing x by ix yields

(1 + x^{2})^{λ−1/2}C_{n}^{λ}(ix) = \frac{(−2)^{n}(λ)_{n}}{n! (n + 2λ)_{n} i^{n}} D^{n} [(1 + x^{2})^{n+λ−1/2}] . (4)

Then

(5) ∫_{−∞}^{∞} i^{n} C_{n}^{λ}(ix) i^{m} C_{m}^{λ}(ix) (1 + x^{2})^{λ−1/2} dx

= \frac{(−2)^{n}(λ)_{n}}{n! (n + 2λ)_{n}} ∫_{−∞}^{∞} \{D^{n}[(1 + x^{2})^{n+λ−1/2}]\} i^{m} C_{m}^{λ}(ix) dx.

Integrating the right hand side of (5) by parts n times, each time differentiating i^{m}C_{m}^{λ}(ix) and integrating the expression in curly brackets, we observe that at the k^{th} stage, the integrated part ±{D^{n−k}(1 + x^{2})^{n+λ−1/2}}D^{k−1}(i^{m}C_{m}^{λ}(ix)) has limit value zero as x tends to ±∞, since λ < −n. After n such integrations by parts we therefore have

(6) ∫_{−∞}^{∞} i^{n} C_{n}^{λ}(ix) i^{m} C_{m}^{λ}(ix) (1 + x^{2})^{λ−1/2} dx

= \frac{(−1)^{n}(−2)^{n}(λ)_{n}}{n! (n + 2λ)_{n}} ∫_{−∞}^{∞}(1 + x^{2})^{n+λ−1/2} D^{n}(i^{m} C_{m}^{λ}(ix)) dx.

If m ≠ n, we may choose n to be larger (or interchange m and n in the preceding argument) and conclude that

∫_{−∞}^{∞} i^{n} C_{n}^{λ}(ix) i^{m} C_{m}^{λ}(ix) (1 + x^{2})^{λ−1/2} dx = 0,

because C_{m}^{λ}(ix) is a polynomial of degree m.

For m = n, the right hand side of (6) is equal to

\frac{2^{n}(λ)_{n}}{n! (n + 2λ)_{n}} ∫_{−∞}^{∞}(1 + x^{2})^{n+λ−1/2} D^{n}(i^{n} C_{n}^{λ}(ix)) dx = g_{n}.

Now (cf. [3, p. 277])

D^{n}[C_{n}^{λ}(x)] = 2^{n}(λ)_{n},

so it follows that

g_{n} = \frac{(−1)^{n}2^{2n}[(λ)_{n}]^{2}}{n! (n + 2λ)_{n}} ∫_{−∞}^{∞}(1 + x^{2})^{n+λ−1/2} dx. (7)

To evaluate the integral, we use the beta integral on a halfline, ([1, p. 7, eq. (1.1.20)])

∫_{0}^{∞} \frac{s^{x−1}}{(1 + s)^{x+y}} ds = \frac{Γ(x) Γ(y)}{Γ(x + y)} :

∫_{−∞}^{∞}(1 + x^{2})^{n+λ−1/2} dx = 2 ∫_{0}^{∞}(1 + x^{2})^{n+λ−1/2} dx

= ∫_{0}^{∞}(1 + t)^{n+λ−1/2} t^{−1/2} dt = \frac{\sqrt{π} Γ(−n − λ)}{Γ(−n − λ + \frac{1}{2})}.

From this the formula for g_{n} is immediate using the duplication formula for the Gamma function.

Corollary. For l, m, n < −λ

∫_{−∞}^{∞} C_{l}^{λ}(ix) C_{m}^{λ}(ix) C_{n}^{λ}(ix) (1 + x^{2})^{λ−1/2} dx

= \frac{(−1)^{s}2^{−2λ−1} (λ)_{s−l} (λ)_{s−n} (λ)_{s−m} (2λ)_{s} (λ)_{l} (λ)_{l+1}[Γ(−l − λ)]^{2}}{(s + λ) (s − l)! (s − n)! (s − m)! (λ)_{s} (2λ)_{2l} Γ(−2l − 2λ)}

where l + m + n = 2s is even and the sum of any two of l, m, n is not less than the third. The integral is zero is all other cases.

Proof. The linearization formula for the product of two Gegenbauer polynomials is given by (cf. [1, p. 319, eq. (6.8.4)])

C_{m}^{λ}(x) C_{n}^{λ}(x) = ∑_{k=0}^{min(m,n)} a(k, m, n) C_{m+n−2k}^{λ}(x) (8)

where

a(k, m, n) = \frac{(m + n + λ − 2k) (λ)_{k} (λ)_{m−k} (λ)_{n−k} (2λ)_{m+n−k} (m + n − 2k)!}{(m + n + λ − k) k! (m − k)! (n − k)! (λ)_{m+n−k} (2λ)_{m+n−2k}}.

From (8) it follows that

(9) ∫_{−∞}^{∞} i^{l} C_{l}^{λ}(ix) i^{m} C_{m}^{λ}(ix) i^{n} C_{n}^{λ}(ix) (1 + x^{2})^{λ−1/2} dx

= ∑_{k=0}^{min(m,n)} a(k, m, n) ∫_{−∞}^{∞} i^{m+n} C_{m+n−2k}^{λ}(ix) i^{l} C_{l}^{λ}(ix) (1 + x^{2})^{λ−1/2} dx

where by Theorem 1 the integral on the right hand side of (9) is zero for l ≠ m+n−2k. Hence only the term with k = s − l = m + n − s survives in the sum, and the right hand side of (9) reduces to

a(m + n − s, m, n) i^{m+n−l} ∫_{−∞}^{∞} [i^{l} C_{l}^{λ}(ix)]^{2} (1 + x^{2})^{λ−1/2} dx

= a(m + n − s, m, n)(−1)^{s−l}g_{l}

= \frac{(l + λ) (λ)_{s−l} (λ)_{s−n} (λ)_{s−m} (2λ)_{s} l!}{(s + λ) (s − l)! (s − n)! (s − m)! (λ)_{s} (2λ)_{l}} \frac{(−1)^{s}2^{−2λ−1}[(λ)_{l}]^{2}[Γ(−l − λ)]^{2}}{l! (l + 2λ)_{l} Γ(−2l − 2λ)} ,

where we have used (1). This is equivalent to the formula given in the corollary.

References

1. G. Andrews, R. Askey, and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, 1999.

2. K. Driver and P. Duren, Zeros of ultraspherical polynomials and the Hilbert–Klein formulas, J. Comput. Appl. Math. 135 (2001), 293–301.

3. E. Rainville, Special functions, Chelsea, New York, 1960.

Kathy Driver, Helmut Prodinger, The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, P. O. Wits, 2050 Johannesburg, South Africa, email: 036kad@cosmos.wits.ac.za,

helmut@gauss.cam.wits.ac.za.