The zeros of basic Bessel functions, the functions Jv+ ax (x), and ...

JOURNAL

OF MATHEMATICAL

ANALYSIS

AND

86, 1-19 (1982)

APPLICATIONS

The Zeros of Basic Bessel Functions, the Functions Jv + Jx), and Associated Orthogonal Polynomials* MOURAD E. H. ISMAIL Department

of Mathematics, Arizona State University, Tempe, Arizona 85287 Submitted by G.-C. Rota

The modified Lommel polynomials are generalized and their orthogonality relation is obtained. As a byproduct we prove that the non-zero roots of J,.+,,(x) are real, simple and interlace with those of J,+.,-,(x). The q-Lommel polynomials are shown to play in the theory of basic Bessel function the role played by the Lommel polynomials in the Bessel function theory. The q-Lommel polynomials are proved to be orthogonal with respect to a purely discrete measure with bounded support. This is then used to prove that the prositive zeros of x-“f”*‘(x; q) are real simple and interlace with the zeros of establish the complete monotonicity of

1. INTR~DUC~~N The Bessel Function

(1.1) satisfies

J,+I(x)=~Jv(x)-J,-I(x).

(1.2)

By iterating the &we recursion one expects that J,+,,,(x) may be expressed as f,(x) J,(x) + g,(x) J,- ,(x), where f,( l/x) and g,( l/x) are polynomials of degrees m and m - 1, respectively. These two polynomials can be expressed in terms of a single set of polynomials called the Pommel polynomials. Indeed J”,,(X) = JL,“WJ”(4

- L1,,+

t(x)J”-,w

(1.3)

* This work is supported in part by National Research Council of Canada Grant A4522 and National Science Foundation Grant MC.5 7903518. 0022-247X/82/030001-19%02.00/0 Copyright T 1982 by Academic Press, Inc. All rights of reproductmn in any form reserved.

2

MOURAD

E. H. ISMAIL

(Watson [26, pp. 294, 2951). These polynomials satisfy the recurrence relation [26, p. 2991

&I-L,(X) + R,+,,,(x) =

2(v

+ m) x R,.,(x)~

m = 1, 2,...,

(1.4)

with Ro,“(x) = 1, R,,,(x) = 2v/x. Following Dickinson [7] we use the notation (h,,,(x)} for what he called the “modified Lommel polynomials,”

h,,(x) = R,,,(llx).

(l-5)

Clearly h,,,(x) is polynomial in x of degree n, when v > -1. It is obvious that

h-l,,(X) + kl+*,, (4 = WV + n) h”,“(X),

n = 1, 2,...,

(1.6)

and

hL”(X)= 19

h, ,Jx) = 2vx.

(1.7)

By a theorem of Favard [9], the recurrence relation (1.6) implies the existence of a positive Bore1 measure dp on (--co, co) so that

with cJ)o = 1,

(a),=a(o+1)**~(a+n-1),

n > 0.

(1.9)

Schwartz [22] computed this measure explicitly. He proved that l(x) is a nondecreasing step function having an increase l/j: _ ,,n at the points fl/j,-l,k, where {j,,,} are the positive zeros of J,(x). It is known that the zeros of x-“J,(x) are real and simple. Dickinson [7] rediscovered Schwartz’ results. A close look at the recursion (1.6) suggests a generalization of the modified Lommel polynomials. Let T,(X) = 5,(x; u, a)

(1.10)

be a sequenceof polynomials defined recursively by

?J(x) = 1, %+1(X) = 2[x(v

SJX) = 2vx + 262.

+ n>+ a]r,(x)-T,-,(X).

(1.11)

THE

ZEROS

OF BASIC

BESSEL

3

FUNCTIONS

Clearly (1.12)

‘“(Xi v, 0) = h”,,(X).

One might expect that the polynomials {t”(x)} will still be connected with Besselfunctions. We shall prove that {r”(x)} are orthogonal with respect to a purely discrete measure. This measure is not symmetric any more because the polynomials {r,(x)} are not symmetric. As a byproduct we prove that J,,+ax(~) has infinitely many zeros for u > -1. The nonzero roots of this function are all real and simple. Furthermore the roots of J,+,,(x) and J v+ux+,t x > int er1ace. It is surprising that these results which do not seem to be directly related to orthogonal polynomials follow from analyzing certain orthogonal polynomials. A direct proof of these results will be of interest. Jackson [ 171 defines a basic or q-analogue of the Besselfunction by Jl”(x;q)=(9;6;;)”

nfo

;

o -1 and a is real. There are infinitely many of them and their onlv cluster point is the point at infinity. Between any two consecutive zeros of x “-“XJC+aX(~)r the function x-“-~-‘-‘J~,+~~- ,(x) has only one zero.

Let (qj} be the sequenceof zeros of x-~-~~-‘J,,+~~-,(x).

‘/,=2~$J~+~~(‘/x)/J.-~+..(l/x)l,=,,.

I

(4.18)

The orthogonality relation for (5,(x; V,a)} is fj

Yjrm(V,'

'; v9 a) r,(V~'

&,,/{2(v

" , v, a) =

+ n)}.

(4.19)

j=O

We conclude by mentioning a basic analogue of r,(x; u, a) and of the functions J r,+ax(x). Define the polynomials (r,&: V,a)} or simply (m,,(x)} by 5, ,p(x; u, a) = 2x(

so,&; v, a) = 1,

1 - 4”) + 2aq”

5n+,,q(~; v, a) = 2[x(l - qn+“) + aq”+“] r,,,(x; v, a) - qn+P-‘tn-,,q(x;

(4.20) v, a).

A generating function is 7

r&x;

ZO

v, a) t” = F

[2tq”(a -x)1”

qk’k-‘)‘* (2(a:x)x)k.

(2xt;

k%O

q)k+

(4*21)

1

Darboux’ method is applicable to (4.21) and leads to (2x)-”

m

r,,,(x; v, a) m r

CO

[(ax)q”/xlk

k(k-I)/2

(&q)k

'

which, in view of Gauss’ binomial theorem (3.4) can be expressedas (4.22)

Let us introduce the notation

WJ (x/2)“+2JqJ(“+J) ,eO

(q;q)J(q”+‘(l

-a);q)J’

(4*23)

18

MoURAD

E. H. ISMAIL

The polynomials of the second kind are ?;,,(x; v, a) = 2( 1 -

qL’) 5,- ,,q(x;

v +

1, a).

(4.24)

The theorems of Favard, Krein and Markoff are’ applicable and if we denote the corresponding measureby &(t), normalize by j?, d/i(t) = 1, then from (4.22), (4.23) and (4.24) we get J.== $$

= 2( 1 -

q”) 3-(

l/x, V,a/q)/cF(

l/x, v - 1,a/q).

(4.25)

This proves that all the non-zero roots of .Y(x; v, a/q) are real and simple and also establishes the interlacing property of these roots for different functions. The orthogonality relation can be similarly found. An explicit formula for r,., can be derived from the generating function (4.21) in the same way the explicit formula (3.6) for h,,,,(x; q) was derived from the generating function (3.3). The result is

j=O

(4;

91,

(4"('

-

alx>i

4)j

(4;

41,

ZJ

(4.26)

ACKNOWLEDGMENTS I am very grateful to Richard Ashey of the University of Wisconsin for suggesting this topic and for several valuable comments. Thanks are also due to Waleed Al-Salam of the University of Alberta for bringing some references to my attention and for his interest.

NOTE

ADDED

IN PROOF

In order to show that x=0 is not a mass point for h,.,(x;q) one needs to show that x:o” (&,,(O; CJ))~diverges, I;.,, being the orthonormal polynomials, This is easy to do since li,,,(x; q) = h,,,(x; q)(l - qo+y q-n+“‘““.‘* and h,,,(O; q) can be computed from the explicit formula (3.6).

REFERENCES

I. N. I. AHIEZER AND M. KREIN, “Some Questions m the Theory of Moments,” Translations of Mathematical Monographs. Vol. 2, Amer. Math. Sot. Providence, R. I.. 1962. 2. G. E. ANDREWS,“The Theory of Partitions,” Encyclopedia of Mathematics and Its Applications, Vol. 2, Addison-Wesley, Reading, Mass., 1977. 3. R. A. ASKEY, The q-gamma and p-beta functions, Applicable Anal. 8 (1978), 125-14 1. 4. R. A. ASKEY AND M. E. H. ISMAIL, A generalization of ultraspherical polynomials, to appear.

THE ZEROS OF BASIC BESSEL FUNCTIONS

19

5. R. A. ASKEY AND M. E. H. ISMAIL, Recurrence relations, continued fractions and

orthogonal polynomials, to appear. 6. T. CHIHARA, “An Introduction to Orthogonal Polynomials,” Gordon & Breach, New York, 1978. 7. D. DICKINSON.On Lommel and Bessel polynomials, Proc. Amer. Math. Sot. 5 (1954). 946-956.

8. A. ERDBLYI, W. MAGNUS,F. OBERHETT~NGER, AND G. F. TRICOM~,“Higher Transcendental Functions,” Vol. 2, McGraw-Hill, New York, 1953. 9. J. FAVARD, Sur les polynBmes de Tchebycheff, C. R. Acad. Sci Paris 200 (1935), 2052-2055.

10. E. GROSSWALD, The student t-distribution of any degreesof freedom is infinitely divisible, Z. Wahrsch. Verw. Gebiefe 26 (1976), 103-109. 11. W. HAHN, Beitrlge zur theorie der Heineschen Reichen, Math. Nachr. 2 (1949), 34&379. 12. M. E. H. ISMAIL AND D. H. KELKER, Special functions, Stieltjes transforms and infinite divisibility, SIAM J. Math. Anal. 10 (1979). 884-901. 13. F. H. JACKSON,On generalized functions of Legendre and Bessel. Trans. Roy. Sot. Edinburgh 41 (1903). l-28. 14. F. H. JACKSON, Theorems relating to a generalization of the Bessel functions, Trans. Roy. Sot. Edinburgh 41 (1903), 105-I 18.

15. F. H. JACKSON,The application of basic numbers to Bessel’s and Legendre’s functions, Proc. London Marh. Sot. (2) 2 (1903-l 904), 192-220. 16. F. H. JACKSON,The application of basic numbers to Bessel’s and Legendre’s functions, second paper, Proc. London Mafh. Sot. (2) 3 (1904-1905), I-20. 17. F. H. JACKSON,The basic gamma function and elliptic functions. Proc. Royal Sot. A 76 (1905), 127-144. 18. J. KENT, Some probabilistic properties of Bessel functions. Ann. Probab. 6 (1978), 166-770.

19. D. MAKI, On constructing distribution functions with applications to Lommel polynomials and Besselsfunctions, Trans. Amer. Math. Sot. 130 (1968), 281-297. 20. F. W. OLVER,“Asymptotics and Special Functions,” Academic Press, New York, 1974. 21. F. POLLACZEK,Sur une gi?n&alisation des polynBmes de Jacobi, M&nor. Sci. Mafh. 131 (1956). 22. H. M. SCHWARTZ,A class of continued fractions, Duke Math. J. 6 (1940), 48-65. 23. L. J. SLATER,“Generalized Hypergeometric Functions,” Cambridge University Press, Cambridge, 1966. 24. M. H. STONE,“Linear Transformations in Hilbert Space and Their Applications to Analysis,” Colloquium Publications, Vol. 15, Amer. Math. Sot., New York, 1932. 25. G. SZEG~,“Orthogonal Polynomials,” Colloquium Publications, vol. 23, fourth ed. Amer. Math. Sot., Providence, R. I., 1975. 26. G. N. WATSON, “A Treatise on the Theory of Bessel Functions,” second ed., Cambridge University Press, Cambridge, 1944. 27. J. WENDEL,Hitting spheres,Ann. Probab. 8 (1980), 164-169.