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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 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.
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.