Georgian Mathematical Journal Volume 11 (2004), Number 3, 409 414 A RECURSION FORMULA FOR THE COEFFICIENTS OF ENTIRE FUNCTIONS SATISFYING AN ODE WITH POLYNOMIAL COEFFICIENTS C. BELINGERI Abstract. A recursion formula for the coefficients of entire functions which are solutions of linear differential equations with polynomial coefficients is derived. Some explicit examples are developed. The Newton sum rules for the powers of zeros of a class of entire functions are constructed in terms of Bell polynomials. 2000 Mathematics Subject Classification: 65L99, 30D20, 30C10. Key words and phrases: Entire solutions of ODE with polynomial coefficients, recursion formulas for coefficients, Newton sum rules for reciprocal of zeros, Bell polynomials. 1. Introduction The problem of finding the Newton sum rules of polynomial solutions of an ordinary differential equation with polynomial coefficients in terms of the coefficients of the same equation was considered in several articles [1], [2], [3], [4], [5]. In particular, in [2] Buendia, Dehesa and Galvez were able to represent the coefficients of the relevant polynomial solutions in terms of the above-mentioned coefficients by proving the following result: Proposition 1.1. Consider the polynomial eigenfunctions P N (x) = const N ( 1) k α k x N k of a linear differential operator of order n g i (x)f (i) (x) = 0, (1.1) where the coefficients g i (x) are polynomials of degree c i, defined by c i g i (x) = a (i) j xj. (1.2) Assume that P N (x) = const N l=1 (x x l), where all x l are different. j=0 ISSN 1072-947X / $8.00 / c Heldermann Verlag www.heldermann.de
410 C. BELINGERI Then the coefficients of P N (x) can be computed recursively in terms of coefficients (1.2) of differential operators by means of the formula: (N s)! s α s (N s i)! a(i) i+q = ( 1) k (N s + k)! α s k (N s + k i)! a(i) i+q k, (1.3) where k=1 q := max{c i i; i = 0, 1,..., n}. (1.4) If c i i (i = 0, 1,..., n), then the differential operator (1.1) is called of hypergeometric type. In the present article the above recursive formula is extended to the case of entire functions satisfying the differential equation (1.1). Some examples are given in Section 3. This gives a possibility to find explicit formulas representing the Newton sum rules of the reciprocal of the zeros of the considered entire function in terms of coefficients (1.2). We do not give these formulas but in the concluding section we give the expression of the Newton sum rules of the reciprocal of the zeros of a class of entire functions in terms of their coefficients using Bell polynomials. 2. Extension of the Recursive Formula We consider now an entire function f(x) = const ( 1) k α k x k (2.1) satisfying the differential equation (1.1), with polynomial coefficients (1.2). Proposition 1.1 can be extended as follows: Proposition 2.1. The coefficients of f(x) are expressed in terms of coefficients (1.2) by means of the recurrent formulas s! s α s (s i)! a(i) i q = ( 1) k (s k)! α s k (s k i)! a(i) i+k q, (2.2) where k=1 q := max{i c i ; i = 0, 1,..., n}. (2.3) Proof. Assuming the normalization such that const = 1, it is sufficient to substitute the derivative f (i) (x) = ( 1) k k! (k i)! α kx k i k=i into equation (1.1), thus obtaining c i ( 1) k a (i) j j=0 k=i α k k! (k i)! xk+j i = 0. (2.4)
A RECURSION FORMULA FOR THE COEFFICIENTS OF ENTIRE FUNCTIONS 411 By using the identity principle of power series, from equation (2.4) we obtain s ( 1) s k (s k)! α s k (s k i)! a(i) i+k q = 0. (2.5) This latter equation gives our result. 3. Examples Example 1. The modified Bessel equation. Consider the modified Bessel equation: x d2 y dx + 3 dy + y = 0. (3.1) 2 2 dx In this equation the coefficients in (1.2) are given by a (2) 2 = 0, a (2) 1 = 1, a (2) a (1) 2 = 0, a (1) 1 = 0, a (1) 0 = 3 2, a (0) 2 = 0, a (0) 1 = 0, a (0) 0 = 1. Furthermore, q = 1. The normalized entire solution is given by Recalling that Γ ( k + 3 2 y(x) = ( 1) k Γ ( k + 3 2 ) = π (2k + 1)!! 2 k+1, we find (3.2) ) xk k!. (3.3) y(x) = 2 ( 1) k 2 k π k!(2k + 1)!! xk. (3.4) Indeed, by using the recurrence formula (2.2) we find α 0 = 1, α 1 = 2 3, α 2 = 2 2 3 5 2!, α 3 = 3 5 7 3!,.... (3.5) Example 2. The Airy equation. Consider the Airy equation d 2 y xy = 0. (3.6) dx2 In this equation the coefficients in (1.2) are given by a (2) 2 = 0, a (2) 1 = 0, a (2) 0 = 1, a (1) 2 = 0, a (1) 1 = 0, a (1) (3.7) a (0) 2 = 0, a (0) 1 = 1, a (0) 0 = 0. Furthermore, q = 2. The normalized entire solution is given by 1 y(x) = 2 3 2k+ 2 3 k! Γ ( ) x 3k. (3.8) k + 2 3 2 3
412 C. BELINGERI Indeed, by using the recurrence formula (2.2) we find α 0 = 1, α 1 = 0, α 2 = 0, 1 α 3 = 3 2 1!, α 4 = 0, α 5 = 0, 1 α 6 = 3 2 2 5 2!, α 7 = 0, α 8 = 0, 1 α 9 = 3 3 2 5 8 3!, α 1 α 11 = 0,.......... Example 3. exponential A Laguerre-type exponential. Consider the Laguerre-type y(x) = e 3 (x) = (see [6]), satisfying the differential equation x k (k!) 4 (3.9) D 3L e 3 (x) := (D + 7xD 2 + 6x 2 D 3 + x 3 D 4 )e 3 (x) = e 3 (x). (3.10) In this equation the coefficients in (1.2) are given by a (4) 4 = 0, a (4) 3 = 1, a (4) 2 = 0, a (4) 1 = 0, a (4) a (3) 4 = 0, a (3) 3 = 0, a (3) 2 = 6, a (3) 1 = 0, a (3) a (2) 4 = 0, a (2) 3 = 0, a (2) 2 = 0, a (2) 1 = 7, a (2) a (1) 4 = 0, a (1) 3 = 0, a (1) 2 = 0, a (1) 1 = 0, a (1) a (0) 4 = 0, a (0) 3 = 0, a (0) 2 = 0, a (0) 1 = 0, a (0) 0 = 1, 0 = 1, 0 = 1. Furthermore, q = 1. Indeed, by using the recurrence formula (2.2) we find α 0 = 1, α 1 = 1, α 2 = 1 (2!) 4, α 3 = 1 (3!) 4, α 4 = 1 (4!) 4,.... 4. Newton Sum Rules and Bell Polynomials (3.11) It is well known that by the Weierstrass factorization theorem an entire function f(z) can be represented in terms of the set of its zeros {z l } l N, z l 0, in the form f(z) = z m e ϕ(z) l=1 ] ( [1 zzl z exp + z2 z l 2zl 2 ) + + zl, lzl l where ϕ(z) is a suitable entire function. Consider now the entire functions which admit a particular factorization ) f(z) = (1 zzl, (4.1) l=1
A RECURSION FORMULA FOR THE COEFFICIENTS OF ENTIRE FUNCTIONS 413 and denote ζ l := 1 z l, l N, so that relation (4.1) becomes f(z) = (1 zζ l ) = 1 σ 1 z + σ 2 z 2 + + ( 1) k σ k z k +, (4.2) l=1 where the variables σ are defined by σ 1 = i ζ i, σ 2 = i<j ζ i ζ j,..., σ k = The relevant Newton sum rules are defined by s 1 = i ζ i, s 2 = i ζ 2 i,..., s k = i i 1 <i 2 < <i k ζ i1 ζ i2 ζ ik,.... ζ k i,... (4.3) (cf. [1]). It is well-known (see, e.g., [7]) that the result related to the following representation formulas of the Newton sum rules by means of Bell polynomials Y k (f 1, g 1 ; f 2, g 2 ;... ; f k, g k ) hold: s k = 1 (k 1)! Y ( ) k 1, σ1 ; 1, 2!σ 2 ;... ; ( 1) k 1 (k 1)!, ( 1) k k!σ k, (4.4) for any k N. It is worth to note that the above notation for the Bell polynomials is not the traditional one, used by Riordan, but it was probably introduced for the first time in [8], showing a link with the Fibonacci and Bernoulli numbers. As it is well-known, the Bell polynomials have wide applications (see, e.g., [9] and references therein, [10]). Therefore, we can conclude that Proposition 4.1. The formula (4.4) gives a representation of the Newton sum rules (4.3) in terms of the coefficients of the Taylor expansion (4.2) of the entire function (4.1). Acknowledgements It is a pleasure to thank Prof. Paolo E. Ricci and the unknown referee for useful comments. This article was partially supported by the research funds of the Ateneo Roma La Sapienza. References 1. K. M. Case, Sum rules for zeros of polynomials. I, II. J. Math. Phys. 21(1980), No. 4, 702 708, 709 714. 2. E. Buendía, J. S. Dehesa, and F. J. Gálvez, The distribution of zeros of the polynomial eigenfunctions of ordinary differential operators of arbitrary order. Orthogonal polynomials and their applications (Segovia, 1986), 222 235, Lecture Notes in Math., 1329, Springer, Berlin, 1988. 3. P. Natalini, Computing the moments of the density of the zeros of classical and semiclassical orthogonal polynomials. (Italian) Calcolo 31(1994), No. 3-4, 127 144 (1996).
414 C. BELINGERI 4. P. Natalini and P. E. Ricci, Computation of Newton sum rules for polynomial solutions of O.D.E. with polynomial coefficients. Riv. Mat. Univ. Parma (6) 3(2000), 69 76 (2001). 5. T. Isoni, P. Natalini, and P. E. Ricci, Symbolic computation of Newton sum rules for the zeros of polynomial eigenfunctions of linear differential operators. In memory of W. Gross. Numer. Algorithms 28(2001), No. 1-4, 215 227. 6. G. Dattoli and P. E. Ricci, Laguerre-type exponentials, and the relevant L-circular and L-hyperbolic functions. Georgian Math. J. 10(2003), No. 3, 481 494. 7. J. Riordan, An introduction to combinatorial analysis. Wiley Publications in Mathematical Statistics. John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. 8. A. Di Cave and P. E. Ricci, Sui polinomi di Bell ed i numeri di Fibonacci e di Bernoulli. Le Matematiche 35(1980), 84 95. 9. P. Natalini and P. E. Ricci, Bell polynomials and some of their applications. Cubo. A Mathematical Journal 5(2003), No. 3, 263 274. 10. Ch. A. Charalambides, On the generalized discrete distributions and the Bell polynomials. Sankhyā Ser. B 39(1977), No. 1, 36 44. Author s address: (Received 17.11.2003; revised 1.06.2004) Università di Roma La Sapienza Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate Via A. Scarpa, 10-00161 Roma Italia E-mail: belingeri@dmmm.uniroma1.it