Buchholz polynomials: a family of polynomials relating
solutions of conuent hypergeometric and Bessel equations
Julio Abad, Javier Sesma∗
Departamento de Fsica Teorica, Facultad de Ciencias, University of Zaragoza 50009 Zaragoza, Spain
Received 7 March 1998; received in revised form 6 July 1998
Abstract
The expansion given by H. Buchholz, that allows one to express the regular conuent hypergeometric functionM(a; b; z) as a series of modied Bessel functions with polynomial coecients, is generalized to any solution of the conuent hypergeometric equation, by using a dierential recurrence obeyed by the Buchholz polynomials. c1999 Elsevier Science B.V. All rights reserved.
AMS classication:33C15; 33C10; 26C05
Keywords:Conuent hypergeometric function; Modied Bessel function; Buchholz polynomials
Special functions, like Bessel and conuent hypergeometric functions, have a wide application in many branches of physics and engineering. Because of this, they have received considerable attention, reected in well-known tables of their properties. Buchholz, in his treatise on the conuent hypergeometric function [4, Section 7, Eq. (16)], gave a convergent expansion of the Whittaker function in series of Bessel functions, namely,
M; =2(z) = (1 +)2z(1+)=2
∞ X
n=0
p() n (z)
J+n(2√z)
(2√z)+n ; (1)
where the p()
n (z) represent polynomials in z2. For the conuent hypergeometric function M(a; b; z) that expansion can be written in the form
M(a; b; z) = (b) ez=22b−1
∞ X
n=0
pn(b; z)Ib−1+n(
√
z(4a−2b))
(√z(4a−2b))b−1+n : (2)
∗Corresponding author. Tel.: + 34 76 76 1265; fax: + 34 76 76 1159; e-mail: [email protected].
Here, the I represent, as usual, the modied Bessel functions and
pn(b; z)≡p(n)(z) with =b−1; (3)
the Buchholz polynomials, dened by the closed contour integral
pn(b; z) =
Expansion (2) has, with respect to other expansions of M(a; b; z) in terms of Bessel functions like, for instance, that given in [3, Eq. 13.3.7], the advantage that the pn(b; z) do not depend on the parameter a. Moreover, they can be written in the form [1]
pn(b; z) =(iz)
as a sum of products of polynomials in b and in z, separately, easily obtainable by means of the recurrence relations, starting with f0(b) = 1 and g0(z) = 1,
where the B2n denote the Bernoulli numbers [3, Table 23.2]. As it can be seen, the coecients of the polynomials are rational and can therefore be managed numerically without error.
In a recent paper [2], related to previous works by Temme [5, 6], we have obtained an asymptotic expansion of the conuent hypergeometric function U(a; b; x) in terms of the asymptotic sequence
(
for large positive values of 2a−b. Comparison of such expansion,
U(a; b; z)∼ 2
with (2) suggests the enunciation of the following:
Proposition 1. Let Zb−1+n represent any linear combination;
Zb−1+n≡Ib−1+n+(−1)nKb−1+n; (10)
of the modied Bessel functions; with coecients and depending arbitrarily on a and b but independent of z. The (at least) formal expansion
satises the conuent hypergeometric dierential equation
The proof, given below, uses a property of the Buchholz polynomials more easily expressed when referred to the reduced Buchholz polynomials, Pn, which we dene as
Pn(b; z)≡z−npn(b; z): (13)
The Pn are polynomials of order n in the variable z, and are even or odd functions of z, according to the index n. Their integral representation, deduced from (4), allows one to obtain immediately the relation
which, starting withP0(b; z) = 1, gives each polynomial in terms of the previous ones. The generating function
can be used to obtain a lot of properties. Here we are interested in the following one.
Lemma 2. The reduced Buchholz polynomials Pn(b; z) obey the recurrent dierential equation
P′
where the primes denote derivation with respect to z.
Proof. The following equations can be obtained from (15):
By using the explicit expression of (b; z; t), it is immediate to check that the sum of the left-hand sides of Eqs. (17)–(21) is identically equal to zero. Therefore, the coecient of tn in the sum of the right-hand sides must vanish.
Incidentally, (16) provides, when integrated, a relation
Pn(b; z) =z
that allows one to obtain each polynomial from the previous one.
Proof of Proposition 1. It is sucient to check that the expansion
E(a; b; z)≡
obeys the dierential equation
zd
Let us denote, for brevity,
y≡pz(4a−2b) and Yb−1+n(y)≡Zb−1+n(y)
yb−1+n : (25)
From the modied Bessel equation [3, Eq. 9.6.1] one deduces that
yd
Substitution of the expansion (23) in the left-hand side of (24) gives, after using (26),
∞
Now, from the recurrence relation [3, Eq. 9.6.26] for the modied Bessel functions, one obtains immediately
ydY(y)
dy =Y−1(y)−2Y(y); (28)
that allows to write (27) in the form
∞
Each term in this sum vanishes identically, according to Eqs. (16) and (13). That proves that Eq. (24) is satised.
= 2b= (a+ 1 −b) in (10), the right-hand side of (11) becomes an asymptotic expansion of
U(a; b; z) for large values of |2a−b|, at least for positive 2a−b and z, as reported in [2]. For nonvanishing values of both and, (11) gives an asymptotic expansion, for large|2a−b|, of the linear combination
1
2b−1 (b)M(a; b; z) +
(a+ 1−b)
2b U(a; b; z) (30)
of the two hypergeometric functions. This follows from the fact that
(
Ib−1+n √x(4a−2b)
√
x(4a−2b)b−1+n )
; n= 0;1;2; : : : ; (31)
form an asymptotic sequence, and the expansion (2), being convergent, becomes obviously asymp-totic for large |2a−b|.
Acknowledgements
The authors beneted from enlightening discussions with J.L. Lopez. The work was nancially supported by Comision Interministerial de Ciencia y Tecnologa.
References
[1] J. Abad, J. Sesma, Computation of the regular conuent hypergeometric function, Math. J. 5 (4) (1995) 74 –76. [2] J. Abad, J. Sesma, A new expansion of the conuent hypergeometric function in terms of modied Bessel functions,
J. Comput. Appl. Math. 78 (1997) 97–101.
[3] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, Dover, New York, 1965. [4] H. Buchholz, The Conuent Hypergeometric Function, Springer, Berlin, 1969.
[5] N.M. Temme, On the expansion of conuent hypergeometric functions in terms of Bessel functions, J. Comput. Appl. Math. 7 (1981) 27–32.
