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Some combinatorial properties of summation operators
JOURNAL
OF COMBINATORIAL
(A) 14,253-255
THEORY
(1973)
Note Some Combinatorial
Properties
of Summation
Operators
NGUYEN-HUU-BONG Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia Communicated by the Managing Editors
Received June 28, 1971 1. NOTATION CHANGES It is clear from the cases considered in my paper [l], that some unifying notation is desirable, for the coefficients of T”+’ (p = 0, 1,2,..., n) in the formula giving Dn(Tk) for all integers k > 1. So we propose making the following changes in notation, all references, unless otherwise indicated, being in [l]: First, for k = 1, instead of (2.5) we have &’
P-l*1
= p&y-l’ + (p - 1) @-1)
withp = 1, 2,..., n + 1 and ug;” u:;
P 2.1
P 1.1
= 0, or equivalently
= (p + 1) aE;l)
+ ptz~-<~\
(4
with p = 0, 1, 2 ,..., n and a:;” = 0. The corresponding changes in the coefficients ur$ (p = 0, 1,2,... n), which appear throughout [l] are now clear. Next, for k > 1, we replace ck+p W) by u:: (0 < p < n) and obtain, instead of (3. lo), a;;
= (p + k) ag,l’
+ (P -
1 + k) a;;:),
which is just (a) when k = 1, noting that uI:;l’ we have D’“‘(Tk)
=
i
(-1)=-P
p=o
for integers k > 0 and 0 < p < n. Copyright All rights
0 1973 by Academic Press, Inc. of reproduction in any form reserved.
253
u$T~+~
,
= 0 and a$”
(i3
= 0. So,
254
NGUYEN-HUU-BONG
2. ADDITIONAL
RESULTS
The relation (3.7) of Theorem 2 can clearly be rewritten as
i. (-‘)“-“p!Y;g = ano for non-negative integer II, where a,, is, as usual, the Kronecker delta. Also, using (3.4) and the results stated in the proof of Corollary 1, we can obtain the following identities, explicitly giving Stirling numbers of the second kind, yon+1 n
-
n@
+
y;‘;
= n(n”
1)
23
(Y)
(n 2 11,
2 1)(3n . 3
-
2)
(n 3 21,
pi-1 = n(n - l>“(fl - 2m + 1) n-2 24 . 3
(n 2 3).
We note that Jordan [2, p. 1711obtained a result equivalent to (y) by different means. Next, considering the case k > 1, the following are immediate consequences of (/3) and of simple inductive arguments a’“’ n-1.k
=
r2 (k + n - 2),-,(2k + n - 1)
(k 2 1; n 2 l),
where we define (x)~ = 1 for x > 0, a’“’ n--2.k
=
n(;3T31) (k + n - 3),-z x {3nz + (12k - 11) n + 2(6k2 - 1% + 5)) (k > 1; n > 2).
If we use these and the relation
$n; = (” - 1 + “) i (-l)“-” P
(: ) (s + k)”
s=O
(k 3 1, 0 < p < n and n > I), which is (3.11) in the modified notation,
SOME COMBINATORIAL
PROPERTIES OF SUMMATION OPERATORS
255
then the following identities, together with a modified version of (3.14), constitute some generalization of part of Corollary 1: i. (-1),-s
(; ) (s + k)p+l = (p ;
l)! (2k + p),
If0 (--lY-”
( ; ) (s + k)p+2 = (p23t: y
{3(p
+
242
+ p}
with integers p > 0 and k 3 1.
REFERENCES
1. N. H. 2.
BONG, Somecombinatorialpropertiesof summation operators,J. Comb&~torial Theory All (1971), 213-221. C. JORDAN, “The Calculus of Finite Differences,” Rottig and Romualter, Budapest;
reprinted by Chelsea, New York, 1960. 3. J. RIORDAN, “Combinatorial Identities,” Wiley, New York, 1968.
OF COMBINATORIAL
(A) 14,253-255
THEORY
(1973)
Note Some Combinatorial
Properties
of Summation
Operators
NGUYEN-HUU-BONG Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia Communicated by the Managing Editors
Received June 28, 1971 1. NOTATION CHANGES It is clear from the cases considered in my paper [l], that some unifying notation is desirable, for the coefficients of T”+’ (p = 0, 1,2,..., n) in the formula giving Dn(Tk) for all integers k > 1. So we propose making the following changes in notation, all references, unless otherwise indicated, being in [l]: First, for k = 1, instead of (2.5) we have &’
P-l*1
= p&y-l’ + (p - 1) @-1)
withp = 1, 2,..., n + 1 and ug;” u:;
P 2.1
P 1.1
= 0, or equivalently
= (p + 1) aE;l)
+ ptz~-<~\
(4
with p = 0, 1, 2 ,..., n and a:;” = 0. The corresponding changes in the coefficients ur$ (p = 0, 1,2,... n), which appear throughout [l] are now clear. Next, for k > 1, we replace ck+p W) by u:: (0 < p < n) and obtain, instead of (3. lo), a;;
= (p + k) ag,l’
+ (P -
1 + k) a;;:),
which is just (a) when k = 1, noting that uI:;l’ we have D’“‘(Tk)
=
i
(-1)=-P
p=o
for integers k > 0 and 0 < p < n. Copyright All rights
0 1973 by Academic Press, Inc. of reproduction in any form reserved.
253
u$T~+~
,
= 0 and a$”
(i3
= 0. So,
254
NGUYEN-HUU-BONG
2. ADDITIONAL
RESULTS
The relation (3.7) of Theorem 2 can clearly be rewritten as
i. (-‘)“-“p!Y;g = ano for non-negative integer II, where a,, is, as usual, the Kronecker delta. Also, using (3.4) and the results stated in the proof of Corollary 1, we can obtain the following identities, explicitly giving Stirling numbers of the second kind, yon+1 n
-
n@
+
y;‘;
= n(n”
1)
23
(Y)
(n 2 11,
2 1)(3n . 3
-
2)
(n 3 21,
pi-1 = n(n - l>“(fl - 2m + 1) n-2 24 . 3
(n 2 3).
We note that Jordan [2, p. 1711obtained a result equivalent to (y) by different means. Next, considering the case k > 1, the following are immediate consequences of (/3) and of simple inductive arguments a’“’ n-1.k
=
r2 (k + n - 2),-,(2k + n - 1)
(k 2 1; n 2 l),
where we define (x)~ = 1 for x > 0, a’“’ n--2.k
=
n(;3T31) (k + n - 3),-z x {3nz + (12k - 11) n + 2(6k2 - 1% + 5)) (k > 1; n > 2).
If we use these and the relation
$n; = (” - 1 + “) i (-l)“-” P
(: ) (s + k)”
s=O
(k 3 1, 0 < p < n and n > I), which is (3.11) in the modified notation,
SOME COMBINATORIAL
PROPERTIES OF SUMMATION OPERATORS
255
then the following identities, together with a modified version of (3.14), constitute some generalization of part of Corollary 1: i. (-1),-s
(; ) (s + k)p+l = (p ;
l)! (2k + p),
If0 (--lY-”
( ; ) (s + k)p+2 = (p23t: y
{3(p
+
242
+ p}
with integers p > 0 and k 3 1.
REFERENCES
1. N. H. 2.
BONG, Somecombinatorialpropertiesof summation operators,J. Comb&~torial Theory All (1971), 213-221. C. JORDAN, “The Calculus of Finite Differences,” Rottig and Romualter, Budapest;
reprinted by Chelsea, New York, 1960. 3. J. RIORDAN, “Combinatorial Identities,” Wiley, New York, 1968.