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Journal of Computational and Applied Mathematics 101 (1999) 237–241

Buchholz polynomials: a family of polynomials relating
solutions of con
uent 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 con
uent hypergeometric function M (a; b; z)
as a series of modied Bessel functions with polynomial coecients, is generalized to any solution of the con
uent
c 1999 Elsevier Science
hypergeometric equation, by using a dierential recurrence obeyed by the Buchholz polynomials.
B.V. All rights reserved.
AMS classication: 33C15; 33C10; 26C05
Keywords: Con
uent hypergeometric function; Modied Bessel function; Buchholz polynomials

Special functions, like Bessel and con
uent hypergeometric functions, have a wide application
in many branches of physics and engineering. Because of this, they have received considerable
attention, re
ected in well-known tables of their properties. Buchholz, in his treatise on the con
uent
hypergeometric function [4, Section 7, Eq. (16)], gave a convergent expansion of the Whittaker
function in series of Bessel functions, namely,
√
∞
X
J+n (2 z)
 (1+)=2
()
√
M; =2 (z ) = (1 + )2 z
pn (z )
;
(1)
(2 z)+n

n=0
where the pn() (z ) represent polynomials in z 2 . For the con
uent hypergeometric function M (a; b; z )
that expansion can be written in the form
√
∞
X
Ib−1+n ( z (4a − 2b))
z=2 b−1
:
(2)
M (a; b; z ) = (b) e 2
pn (b; z ) √
( z (4a − 2b))b−1+n
n=0
∗

Corresponding author. Tel.: + 34 76 76 1265; fax: + 34 76 76 1159; e-mail: javier@posta.unizar.es.

c 1999 Elsevier Science B.V. All rights reserved.

0377-0427/99/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 9 ) 0 0 2 2 6 - X

238

J. Abad, J. Sesma / Journal of Computational and Applied Mathematics 101 (1999) 237–241

Here, the I represent, as usual, the modied Bessel functions and
pn (b; z ) ≡ pn() (z )

with  = b − 1;

(3)

the Buchholz polynomials, dened by the closed contour integral
(−z )n
pn (b; z ) =
2i

Z

(0+)

z
1
exp −
coth t −
2
t




 

sinh t
t

b−2

1
d t:
t n+1

(4)

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]
[n=2]
(iz )n X n
fs (b) gn−2s (z );
pn (b; z ) =
n! s=0 2s

!

(5)

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,
fs (b) = −



X
s−1
2s − 1 22(s−r) |B2(s−r) |
b
−1
fr (b);
2
s−r
2r
r=0

iz
gm (z ) = −
4

!

[(m−1)=2]

X
k=0

!

m − 1 22(k+1) |B2(k+1) |
gm−2k−1 (z );
2k
k +1

(6)

(7)

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 con
uent hypergeometric function U (a; b; x) in terms of the asymptotic sequence
)
(
√
Kb−1+n ( x(4a − 2b))
√
; n = 0; 1; 2; : : : ;
(8)
( x(4a − 2b))b−1+n
for large positive values of 2a − b. Comparison of such expansion,
√
∞
X
Kb−1+n ( (4a − 2b)z )
2b e z=2
n
(−1) pn (b; z ) √
;

U (a; b; z ) ∼
(a + 1 − b) n=0
( (4a − 2b)z )b−1+n

(9)

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)n Kb−1+n ;

(10)

of the modied Bessel functions; with coecients  and  depending arbitrarily on a and b but
independent of z . The (at least) formal expansion
√
∞
X
Zb−1+n ( z (4a − 2b))
z=2
(11)

w (z ) = e
pn (b; z ) √
( z (4a − 2b)) b−1+n
n=0

239

J. Abad, J. Sesma / Journal of Computational and Applied Mathematics 101 (1999) 237–241

satises the con
uent hypergeometric dierential equation
z

d2 w
dw
− aw = 0:
+ (b − z )
dz2
dz

(12)

The proof, given below, uses a property of the Buchholz polynomials more easily expressed when
referred to the reduced Buchholz polynomials, Pn , which we dene as
Pn (b; z ) ≡ z −n pn (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
[(n+1)=2]
[n=2] 2k
X
2 B2k
z X
22k B2k
(2k − 1)
Pn−2k+1 (b; z ) + (b − 2)
Pn−2k (b; z );
nPn (b; z ) =
2 k=1
(2k )!
(2k )!
k=1

(14)

which, starting with P0 (b; z ) = 1, gives each polynomial in terms of the previous ones. The generating
function
1
z
coth t −
(b; z; t ) ≡ exp
2
t
 

 

sinh t
t

b−2

=

∞
X

Pn (b; z ) t n

(15)

n=0

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
Pn′ +

1
b−2 ′
n
′′
Pn−1 − Pn−1
;
Pn = Pn−1 +
2z
4
z

(16)

where the primes denote derivation with respect to z .
Proof. The following equations can be obtained from (15):
∞
d (b; z; t ) X
=
Pn′ (b; z ) t n ;
dz
n=1
∞
n
t d (b; z; t ) X
Pn (b; z ) t n ;
=
2z
dt
2
z
n=1


∞ 
X
1
1
− t(b; z; t ) =
Pn−1 (b; z )t n ;
−
4
4
n=1

−
t


∞ 
b−2
b − 2 d (b; z; t ) X
′
=
Pn−1
(b; z ) t n ;
−
t
z
dz
z
n=1

∞
d 2 (b; z; t ) X
′′
=
(b; z ) t n :
Pn−1
dz2
n=1

(17)

(18)

(19)

(20)

(21)

240

J. Abad, J. Sesma / Journal of Computational and Applied Mathematics 101 (1999) 237–241

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 t n in the sum of
the right-hand sides must vanish.
Incidentally, (16) provides, when integrated, a relation
Pn (b; z ) = z −n=2

Z z
0

1
b−2 ′
′′
(b; v) vn=2 d v;
Pn−1 (b; v) +
P (b; v) − Pn−1
4
v n−1


(22)

that allows one to obtain each polynomial from the previous one.
Proof of Proposition 1. It is sucient to check that the expansion
√
∞
X
Zb−1+n ( z (4a − 2b))
E (a; b; z ) ≡
pn (b; z ) √
( z (4a − 2b)) b−1+n
n=0

(23)

obeys the dierential equation
z

dE
b z
d2 E
− a− +
+b
E = 0:
2
dz
dz
2 4




(24)

Let us denote, for brevity,
y≡

p

z (4a − 2b)

and

Yb−1+n (y) ≡

Zb−1+n (y)
:
y b−1+n

(25)

From the modied Bessel equation [3, Eq. 9.6.1] one deduces that
y

d 2 Y
d Y
− yY = 0:
+ (2 + 1)
dy2
dy

(26)

Substitution of the expansion (23) in the left-hand side of (24) gives, after using (26),
∞ 
X
n=0

zpn′′

+

bpn′

z
d Yn (y)
n
− pn Yn (y) + pn′ − pn y
:
4
2z
dy








(27)

Now, from the recurrence relation [3, Eq. 9.6.26] for the modied Bessel functions, one obtains
immediately
y

d Y (y)
= Y−1 (y) − 2Y (y);
dy

(28)

that allows to write (27) in the form
∞ 
X
n=0

zpn′′ + (b − 2n )pn′ −



z
n+1
nn
′
pn+1 Yn :
−
−
pn + pn+1
4
z
2z




(29)

Each term in this sum vanishes identically, according to Eqs. (16) and (13). That proves that
Eq. (24) is satised.
If in (10) we choose  = 0, the expansion (11) turns out to be convergent and, for  = 2b−1 (b),
coincides with M (a; b; z ), as we had mentioned at the beginning of this letter. For  = 0 and

J. Abad, J. Sesma / Journal of Computational and Applied Mathematics 101 (1999) 237–241

241

 = 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


( a + 1 − b)
1
M (a; b; z ) + 
U (a; b; z )
2b−1 (b)
2b

of the two hypergeometric functions. This follows from the fact that
(

)
√
Ib−1+n x(4a − 2b)

b−1+n ; n = 0; 1; 2; : : : ;
√
x(4a − 2b)

(30)

(31)

form an asymptotic sequence, and the expansion (2), being convergent, becomes obviously asymptotic for large |2a − b|.
Acknowledgements

The authors beneted 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 con
uent hypergeometric function, Math. J. 5 (4) (1995) 74 –76.
[2] J. Abad, J. Sesma, A new expansion of the con
uent hypergeometric function in terms of modied 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 Con
uent Hypergeometric Function, Springer, Berlin, 1969.
[5] N.M. Temme, On the expansion of con
uent hypergeometric functions in terms of Bessel functions, J. Comput. Appl.
Math. 7 (1981) 27–32.
[6] N.M. Temme, Uniform asymptotic expansions of a class of integrals in terms of modied Bessel functions, with
application to con
uent hypergeometric functions, SIAM J. Math. Anal. 21 (1990) 241–261.