Solution to Problem 2 in the SIAM Activity Group on Orthogonal Polynomials and Special Functions Newsletter

by Richard Askey and George Gasper (askey@math.wisc.edu and george@math.nwu.edu)

September 5, 1997 version

Problem 2

Is it true that

x^{2}t^{x}_{2}F_{1}(x + 1, x + 1; 2; 1 − t)

is a convex function of x whenever −∞ < x < ∞ and 0 < t < 1?

Submitted by George Gasper, August 19, 1992.

Solution to Problem 2 by Richard Askey and George Gasper

(askey@math.wisc.edu and george@math.nwu.edu)

Let 0 < t < 1, −∞ < x < ∞, and set

f_{t}(x) = x^{2}t^{x}_{2}F_{1}(x + 1, x + 1; 2; 1 − t).  (1)

By using the binomial theorem to expand t^{x} = (1−(1−t))^{x} in powers of 1 − t, it follows that

f_{t}(x) = x^{2} ∑_{j=0}^{∞} ∑_{k=0}^{∞} \frac{(−x)_{j}(1 + x)_{k}(1 + x)_{k}}{j! k! (2)_{k}} (1 − t)^{j+k}

= x^{2} ∑_{n=0}^{∞} \frac{(−x)_{n}}{n!} (1 − t)^{n} _{3}F_{2}(−n, 1 + x, 1 + x; 2, 1 + x − n; 1)

after setting j = n − k and changing the order of summation. Then, application of the b = −n case of the transformation formula [1, 3.8(1)] to the above _{3}F_{2} series yields the expansion formula (generating function)

f_{t}(x) = x^{2} ∑_{n=0}^{∞} F_{n}(x) (1 − t)^{n}  (2)

with

F_{n}(x) = _{3}F_{2}(−n, 1 + x, 1 − x; 2, 1; 1), n = 0, 1, …. (3)

From (3) it is clear that each F_{n}(x) is a polynomial of degree n in x^{2}, and hence, by (2), that f_{t}(x) is an even function of x. Also, computations of the coefficients of the polynomials F_{n}(x) for many values of n suggest that each F_{n}(x) is an absolutely monotonic function (one whose power series coefficients are nonnegative) of x. Then (2) would imply that f_{t}(x) is an absolutely monotonic function of x and, consequently, a convex function of x when 0 < t < 1. Thus it suffices to prove that each F_{n}(x) is an absolutely monotonic function of x.

First observe that

F_{n}(x) = \frac{1}{n! (n + 1)!} S_{n}(−x^{2}; 1, 1, 0),  (4)

where

\frac{S_{n}(x^{2}; a, b, c)}{(a + b)_{n}(a + c)_{n}} = _{3}F_{2}(−n, a + ix, a − ix; a + b, a + c; 1),

(5)

n=0,1,…, are the continuous dual Hahn polynomials, see [4, §1.3]. Since the polynomials {S_{n}(y; a, b, c)}_{n≥0} are orthogonal on the interval (0, ∞) with respect to a positive weight function when c ≥ 0 and either a, b > 0 or a, b are complex conjugates with positive real parts, the zeros y_{k,n}(a, b, c), 1 ≤ k ≤ n, of each S_{n}(y; a, b, c) are positive, and hence

S_{n}(x^{2}; a, b, c) = C_{n}(a, b, c) ∏_{k=1}^{n} (y_{k,n}(a, b, c) − x^{2})

where, by inspection of the right side of (5), C_{n}(a, b, c) > 0. Therefore, with the above-mentioned restrictions on a, b, c each

S_{n}(−x^{2}; a, b, c) = C_{n}(a, b, c) ∏_{k=1}^{n} (y_{k,n}(a, b, c) + x^{2}) (6)

is an absolutely monotonic function of x, and hence, by (4) and (2), F_{n}(x) and f_{t}(x) are absolutely monotonic functions of x, which completes the proof. From this proof it is clear that the function t^{x}_{2}F_{1}(x + 1, x + 1; 2; 1 − t) is also an absolutely monotonic, and hence convex, function of x. For a discussion of the origin of this problem, see page 11 of the Fall 1994 issue of this Newsletter (note that in the fourth displayed equation (K_{i(x+iy)}(a))^{2} should read (K_{ix}(a))^{2} ). Remark 1. Formula (2) can also be derived by first applying the Pfaff-Kummer transformation formula [3, (1.5.5)] to the _{2}F_{1} series in (1) with the additional restriction that |(1 − t)/t| < 1, using the binomial theorem to expand each of the t^{−j−1} powers of t in powers of 1 − t, changing the order of summation, and then employing analytic continuation to remove the |(1 − t)/t| < 1 restriction. Remark 2. From (6) and an extension of the derivation of (2) it follows that the generating function (cf. [4, (1.3.6)], [5, p. 398])

(1 − t)^{x−c} _{2}F_{1}(a + x, b + x; a + b; t)

= ∑_{n=0}^{∞} \frac{S_{n}(−x^{2}; a, b, c)}{n! (a + b)_{n}} t^{n}, 0 ≤ t < 1,

is an absolutely monotonic function of x when c ≥ 0 and either a, b > 0 or a, b are complex conjugates with positive real parts, and that the generating function (cf. [2, 19.10 (25)], [4, (1.3.9)])

e^{t} _{2}F_{2}(a + x, a − x; a + b, a + c; −t)

= ∑_{n=0}^{∞} \frac{S_{n}(−x^{2}; a, b, c)}{n! (a + b)_{n} (a + c)_{n}} t^{n}, 0 ≤ t < ∞,

is an absolutely monotonic function of x when a, b, c > 0. Remark 3. Similarly, additional absolutely monotonic functions can be obtained by using the generating functions for the Wilson W_{n}(x^{2}; a, b, c, d) and the Askey-Wilson p_{n}(x; a, b, c, d|q) polynomials in [4, §1.1 and §3.1], and their limit cases, along with suitable changes in variables. In particular, it follows from [4, (1.1.2) and (1.1.6)], [6, (1.2) and (2.4)] that the generating function

2F_{1}(a + x, b + x; a + b; t)_{2}F_{1}(c − x, d − x; c + d; t)

= ∑_{n=0}^{∞} \frac{W_{n}(−x^{2}; a, b, c, d)}{n! (a + b)_{n} (c + d)_{n}} t^{n}, 0 ≤ t < 1,

is an absolutely monotonic function of x when Re(a, b, c, d) > 0 and non-real parameters occur in conjugate pairs with a + b > 0 and c + d > 0.

References

[1] Bailey, W.N.: Generalized Hypergeometric Series, Cambridge University Press, London, 1935: reprinted by Stechert-Hafner Service Agency, New York–London, 1964.

[2] Erdélyi, A. et al.: Higher Transcendental Functions, Vol. III, McGraw-Hill, New York–London, 1955.

[3] Gasper, G. and Rahman, M.: Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications, Vol. 35, Cambridge University Press, 1990.

[4] Koekoek, R. and Swarttouw, R.F.: The Askey-scheme of hypergeometric orthogonal polynomials and its qanalogue, Delft University of Technology, Faculty of Technical Mathematics and Informatics, Report 94-05, 1994; revised version of February 20, 1996 obtainable by anonymous ftp from: unvie6.un.or.at, directory siam/opsf new/koekoek swarttouw, file koekoek swarttouw1.ps, or via the World Wide Web:

ftp://unvie6.un.or.at/siam/opsf new/

koekoek swarttouw/koekoek swarttouw1.ps

[5] Letessier, J. and Valent, G.: Dual birth and death processes and orthogonal polynomials, SIAM J. Math. Anal. 46 (1986), 393–405.

[6] Wilson, J.A.: Asymptotics for the _{4}F_{3} polynomials, J. Approx. Theory. 66 (1991), 58–71.