The complete Bell polynomials for certain arguments in terms of Stirling numbers of the first kind

The complete Bell polynomials for certain arguments in terms of Stirling numbers of the first kind

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS Journal of Computational and Applied Mathematics 51 ( 1994) 113-l 16 Letter Section The complete ...

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS Journal of Computational

and Applied Mathematics

51 ( 1994) 113-l 16

Letter Section

The complete Bell polynomials for certain arguments in terms of Stirling numbers of the first kind K.S. KGlbig * Computing

& Networks Division, CERN, CH-1211 Gentve 23, Switzerland

Received 28 September

1993; revised 17 December

1993

Abstract

The value of the (exponential) complete Bell polynomials for certain arguments, given essentially by finite sums of reciprocal powers with a real parameter, is expressed in terms of coefficients generated by the power series expansion of a Pochhammer symbol. This result generalizes an earlier result given by Comtet, relates these polynomials for certain arguments to Stirling numbers of the first kind. Keywords: Complete

Bell polynomials;

Stirling numbers of the first kind; Pochhammer

which

symbol

1. Introduction When evaluating certain infinite integrals with powers of logarithms by repeated differentiation gamma functions, e.g., in [ 31, &(a,

Y) =

Jrn 0

e-“‘t”-’ log”* t dt,

Rea > 0,

of

Re Y > 0, m E No,

for integers v = FZ,n E N, one soon realizes that the Stirling numbers of the first kind Sf), defined by [ 1, No. 24.1.31 log’(1 +x>

=j!

~S~)$, h=j

can be expressed

h < j, .

by polynomials

in finite sums over reciprocal

* e-mail: [email protected]. 0377-0427/94/$I7.00 @ 1994 Elsevier Science B.V. All rights reserved SSDIO377-0427(94)00010-X

powers of integers

114

K.S. Kiilbig/Journal

Examples

of Computational

and Applied

Mathematics

51 (1994)

113-I

16

of such relations are

sC2) l), II+1= -(-l)“n!fl(lt, sC3) {&n, 1) - g(n,2)}, II+1= i(-l)“n! $4) n+l

-

-i(-l)“n!

{a3(n, 1) - 3~(n,

(1) l)a(n,2)

+2a(n,3)}.

These relations are not obvious whether from the well-known recurrence relation [ 1, No. 24.1.31 or from the explicit Schlomilch formula [2, p.2161, [4] for Sy). Similarly, considering R, (a, V) for half-integers v = II + i, one finds that certain expressions involving Stirling numbers of the first kind are related to polynomials in finite sums over reciprocal powers of the odd integers

for example,

&) k=2

ks’k’ n

0

k 22-k = (-1)” 2

1) -G(n,2)}.

(2)

A theorem given by Comtet [ 2, p.2171 explains the equations ( 1). This theorem has been extended in [4], in order to cope with relations of type (2). It is the purpose of this note to generalize the result of Comtet further, thus giving a connection between the value of the complete Bell polynomials for arguments composed essentially of sums of reciprocal powers with a real parameter, and the coefficients generated by the power series expansion of a Pochhammer symbol, which in turn are linear combinations of Stirling numbers of the first kind.

2. The Theorem Theorem. p.1341

Let Y,( XI,...,

\

x,, ) be the pth (exponential)

tx

complete

Bell polynomial,

dejined by [ 2,

tP

and .S$i) the Stirling numbers of the first kind. Let further Q E E-X,(Y # - 1, -2, . . . , -9,

and v&5,527..

-3 s,>

=~,((51.-1!52,...,(-l)p-'(p-

lMp).

K.S. Ktilbig/Journal

of Computational

and Applied Mathematics

51 (1994) 113-116

115

Then, for p, 9 E N, (3) where S(p,q;a)

= k(-l)‘+“s::’ .i=i,

are the coejficients

; (I

(1 + @>‘-a

in the expansion of the Pochhammer symbol [ 1, No. 6.1.221 (4)

(1 + Ly+ X>(, = e S(p, 9; a)x” p=o in powers

of x.

Proof. We first show that relation (4) holds. From a formula in [ 1, No. 24.1.31 it is easily seen that, with (x)~ = 1 and $0) = S,, the Pochhammer symbol ( x)~ can be written as (x),

= x(x + 1)

. . .(x+9_

1) =~(-l)jtqs:ij)xj.

(5)

j=O

Therefore, (1 + CY+ x)q = 2(-l

).i+qSf) 2

;

(1 + ,)j-px”

I

(1 + C#-Px”

il

p=o

j=O

Y = i-;k(_ljj+,,:,il

IF0 .j=11

0

= c S(p, q; a)xP. P=o

We now make use of the well-known fact that, for a function f(x) the pth derivative of exp f( x) is given by [ 5, p.351 Kef(x) dxi’

= ef(“)$((f’(x),

f”(x),

. . . , f(P)(x)).

Thus, with f(x)

= log( 1 + 0~+ x)q = e log(r + my+ x) I=l

and fCk’(x)

= (-l)k-’

(k-

l)!s(q,k;a+x),

ka

1,

we obtain $1

+a+x),

= (1 +cu),V,(s(q,l;a),s(9,2;(~),...,~(9,p;a)). X=0

which is p times differentiable,

116

K.S. Kolbig/Journal

Equation

of Computational and Applied Mathematics 51 (1994) 113-116

(3) follows with the help of (4).

Because of ( x)~+, = x( 1 +x),, S(p,q;O)

= &-])j+?Slj) j=p

0

we have for (Y= 0 from (4) and (5) j

= (-])/‘+4$;~‘),

0P

and hence

p+qp?p+l),

vp(dq,lLm,2)

v...,

dq,p))=

(-I) { 0,

q!

q+l

P G 41 P > 4.

This is the result given by Comtet, proved in a different way by using the power series expansion of log(1 +x>. By setting (Y= -i, p = 2, q = n in (3), and noting that s( q, k; -i) = 2%( q, k), we obtain (2) as another special case. The polynomials V, (& , . . . , tp) can be calculated from I$ (x1, . . . , x,,) [ 2, p.3071 and the first five are given by

We may add that, in a recent Editorial comment on a solution of a problem posed by Wepsic [6], it is noted that Sy’ can be expressed by a polynomial in (T( q, k). The first four polynomials are given explicitly. There is, however, no hint as to their relation to the complete Bell v,(&..&) polynomials YP( x1, . . . , xp) . References [ l] M. Abramowitz and LA. Stegun, Eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 9th printing with corrections, 1972). [ 21 L. Comtet, Advanced Combinatorics (Reidel, Dordrecht, 1974) _ [3] K.S. KBlbig, On the integral &m e-r“t’-’ log”’ tdt, Math. Comp. 41 (1983) 171-182. [4] K.S. Kdlbig, On the integral so” nY-’ ( 1 + fix)-” In” xdx, J. Comput. Appl. Math. 14 (3) ( 1986) 319-344. [ 51 J. Riordan, An Introduction to Combinatorial Analysis (Wiley, New York, 1958). [6] E. Wepsic, Problem E3412. Balls and urns at random, Amer. Math. Monthly 100 (1993) 81-84.