ON THE SUMS XXo¿p AND JZi-o(-l)"kp
DAVID ZEITLIN
1. Introduction. Let uk satisfy a linear, difference equation of order
m with real, constant coefficients:
um+k + aium+k-i
+ a2um+k-2 +
• • • + amuk = 0,
ßo = 1, am 9e 0, k = 0, 1, • • ■ .
Using our formula, (2.1), for Et-o u^k, we will give a closed form for
zZt-oukrk, as well as closed forms for XXo &" and E"=o ( — l)kkp,
which are also expressible in closed form by means of Bernoulli and
Euler polynomials of the first order, respectively. We will show that
n
-2p+lJZ(-l)kk"
(1-2)
=Ê2(-lr(
P+L\)[(-l)^(n+l+s)p-sp]
k=o «-o
\s+k+l/
(P = 0,1, • • • ),
"
p
*=o
*=o
/n+p+l-kXÇi,
/ p+l
\
\s+k+l/
p + I
/ B-0
\
(p - 0, 1, - • . ).
Closed forms for Et-o kpukrk, E?-o M*r*cos (&#)>and E"-o «*''* sm (&?)
may also be obtained from (2.1), but the details are omitted.
2. A lemma.
We now derive the following result:
Lemma 1. Let uk satisfy (1.1). Then
1 + JZ of1
Proof
(2.2)
T,ukrk =¿Z\
X) ajUnJrm-k-i rn+m'k
m-1 r- m-l-t
-]
¡fc-oL y-o
J
of (2.1). Let Ait i = l, 2, • • • ,m,he
real constants,
»„+Í + Aivm+k-i + A2vm+k-2+ ■ ■ ■ + Amvk = 0,
Defining Avk= Vk+i—vk,(2.2) can be rewritten
and let
Am ^ 0.
as
Received by the editors June 1962; and, in revised form, April 15, 1963.
642
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ON THE SUMSYl-o
(-I)**»
A?;
'm+4-2-
1 + Y Ai
y+i
(2.3) Aflm+*_i+ Y
moreover,
l + Y^i
¡—i
by summing
643
*" AND^=0
-/ +
i>k= 0;
both sides of (2.3) for k = 0, 1, • • • , », we
obtain
m-2 r
(»n+m— »to-i) + Eh+E^'
y-oL
(2.4)
j+1
-1
»-i
J
(Vn+m-l-j ~ »m-2-j)
î + XM,Y% = 0.
i=l
If we set Vk= ukrk, then Ai=a¡r\
m-2
J *=0
* = 1, 2, • • • , m. Since
y+1
X Y a¿rn+m_1_ Un+m—1—>
j=0 i=l
TO—2 m—l—k
Jt=0
(2.5)
j-1
n—1 pm—1—A
"1
^
Y ajUn+m-k-i rn+m-k
k=0 L j=0
J
_
£
«„+m_tr«+»»-*)
(2.4) simplifies to (2.1).
3. Application of Lemma 1. The generating function of uk is readily
obtained from (2.1). Let R>0 and suppose 2^L0 M*r*converges for
|r| <R. Then, for \r\ <R, unrn^>0 as »->°°.
gives, as »—>=o,
p
(3.1)
m
"]
oo
Thus,
for |r| <i?,
(2.1)
m—1 ["m— 1—&
1 + Y «S* Y »*rk= Y \ Y aium-i-k-i r"-1"*.
L
<-i
J *=o
k=oL y-o
J
Another proof of (3.1) (based on the additive property of generating
functions), is given in Jordan [l, p. 27]. We note that the generating
function of uk is a rational function in r.
Recalling the well-known identity
Ci
ÍP + i\
l~o
(s - 0,1, • - - , p)
.(¿+l)!(-l)p+1
(s = p + l)
it follows that Uk= kp, p = 0, 1, • • • , satisfies
*+i
(p + 1\
,_o
\
Y (-i)'í
Thus,
from (2.1), with
;
. K+1+*-/= 0.
/
uk= kv, m = p + l, and
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]•
644
DAVID ZEITLIN
[August
j = 0, 1, • • • , p-\-lt we obtain
n
(-l)p(r-
1)"+1X) ¿^
V-k
=E
(3.3)
p
-E
/A
_|_ 1\
E(-l)^
.
r v-k
/j) _i_ j\
£(-!)'(
k=o L y=o
-1
)(n + p+l-k-j)p\
. )(p-k-j)p
\
J
«n+p+1—k
r»-».
/
If we set r= —1 in (3.3), we obtain (1.2).
Let/(r) denote the right-hand side of (3.3) so that
71
(-l)p(r-
l)p+lJZkprk =f(r).
k=0
In §4, it will be shown that [/(8,W]r=i = 0 for 5 = 0, 1, • • • , p. Thus,
applying
L'Hospital's
rule (£ + 1) times to f(r)/(r
—l)p+1 for r= 1, we
obtain (1.3) from (3.3), noting that
(n + p+l-kyp+»
= (p+l)\(
(n + p + 1 - k\
*
),
\
p+ I
/
where x(-p+1)=x(x—l) • ■ • (x —p).
Remarks.
Nielsen [2, p. 27, (8)] defines for non-negative
integers
M, N, and P
M,N
(3.4)
*
(M
BP ,(x) = JZ(-iy(
+ l\
.
)(x+p-j)».
Since
p—A;
P-*
/
s=o
V
+ ¿k ++ 1/
<5 +
E(-l)s
* +
1
; , ,
\
)(n+l
+ s)p
= (-l)^E(-lV(
(1.3) can be rewritten
(3.5) E^
¡fc-l
. )(n+l
+ p-k-j)p,
as
= (-DpE(
»-0 \
^
# + 1
)^,P(» + 1) (#-1,2,...).
/
Noting that 5^x(x) = 0 (see (3.2)), we obtain from [2, p. 237, (5),
with n and £ interchanged,
x = n + l, m = p]
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i964]
tkp=tï(2n+i+S)-(n+1
(3.6)
645
ON THE SUMSZ*-o ** ANDX"-o (- D**"
ti
ir0La p +1
+ S\'
I
B7(n+l)
\ p + i ).
(P = 1, 2, • • • ).
Combining (3.5) and (3.6), we obtain
r
. "
* (2n
[i + (-Dp]2>
*=1
(3.7)
+ 1 + s\
=E
,=0 \
P + 1
/
+D
(P - 1, 2, • • • ).
Thus, from (3.7), we conclude
(3-8)
Z
that
,/2» + 1 + 5
,_o \
' : ; ~)2£~0»+ 1)-0
p + 1
"
.p
jSf'Cn+l)
ft=l
s=0\
4. Additional
(#-1,3,5,-
/
p /2» + 1 + A
(3.9) 2l>=Z
P +
1
•),
(# = 2,4,6,..-).
/
proofs. We will require the following result:
Lemma 2. For s = 0, 1, • • ■ , p, and arbitrary
p / #\
(4.1)
PtP
J_1 K(»
» /x + # - k\ Ç*
L(
VZl
¿=o \t /
«-o \
x, we have
(p + 1\
)Z(-i)> \ J. /)(#-*-i)'-1
/ y-o
í
= A"-s1p,
wAere
A-l»= E (-l)r(
(4.2)
0
)(»-r+l)».
Proof of (4.1). Let U(x, s) denote the left-hand side of (4.1). Since
we have
t(
X )u(x,s)
,=o \p — s/
(4.3)
,=0 \ 1 /
4=0 \
P
/ j=0
\
J
/
^É(>É(2*+0£c->)f+W<.
<«o\l
/
k=o\
P
I y=o
v J
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I
646
DAVIDZEITLIN
[August
Nielsen [2, p. 28, (10) and (14)] has shown that
....
(4.4)
,
„
™ (y + k - l\
(-D—y = i_o\
Z(
k
/m+l\
)E(-i)'( \ J. /)(*-i)"
) ,-_„
m
(m 2: n).
Setting m = p, n = p—i, and y —2x+l
(4.5) JZ(
in (4.4), we obtain from (4.3)
)u(x,s)=iz(P)(-x)i(2x+l)p-i=(x+l)p.
,=o \P — s/
¿_o\ t /
Since the expansion,
(x+l)p= Jz(x)a'Ip,
,_o \ 5 /
known as Newton's
interpolation
formula,
is unique
(see [3, p. 27],
[4, p. 17]), it follows that U(x,p-s) sA'li», and thus U(x, s) =Ap-"1¡\
This completes the proof of (4.1).
In §3, we claimed that |/(*)(*)]¡e-i=/(*)(l) =0 for 5 = 0, 1, • • • , p.
To show this, \etf(x)=h(x)—g(x),
p r p-k
where (see (3.3))
/í4-i\
1
i+p+i-t
(4.6)*(*)= E I E (-l)M . J (»+ # + 1- *-j)'\ xn
(4.7)g(x)= èr2(-i)yfi+1)f>-*-i)*i*^.
We now proceed to show that h(,)(I) = gU)(I) =s\Ap~'lp,
• • • , p. Using the binomial
theorem
5 = 0, 1,
and noting that
(n + p + 1 - k\
= s\(
r
\,
(n + p+l-k)^
we obtain from (4.6)
A<«>(1)
= JZ{n+p+l-k)^JZ(-iy(P+V)(n+p+l-k-j)p
k=0
(4.8)
y=0
\
J
/
= s<.±(p)(»+1y±("+p+i-h)
<_o\ î /
k=o\
p-k
/p+l\
y-o
\ J
s
I
• E (-1)' ( . Kp-h-f)*-' = í!A«~l>,
/
having applied (4.1) with x = n + l.
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1964]
ON THE SUMSE*-o k" AND Et-o
(-1)1*"
647
If we set x = 0 in (4.1), the only nonzero term occurs for i = 0. Thus,
(4.9)
,?, IP —k\ tí
/* + 1\
k-0\
\
E(
)L(-1)M
S
/ y=o
. )(p-k-j)>
J
= A>~l>
/
(5 = 0, 1, • • •,/»);
and from (4.7), we have
g(>,(i)=s\±(p~
(4.10)
k)Y(-w(p
*=A s Ji~°
+ 1)(p-k-j)'
Kj J
- jIap-'I».
Thus,/">(l)E=A<"(l)-g(,)(l)=0,
5 = 0, 1, •••,#,
as claimed.
Acknowledgment. I wish to thank Professor L. Carlitz for providing
me with the outline of the proof for (4.1).
References
1. C. Jordan, Calculus of finite differences, Chelsea, New York, 1960.
2. N. Nielsen, Traité élémentaire des nombres de Bernoulli, Gauthier-Villars,
Paris,
1923.
3. L. M. Milne-Thomson,
The calculus of finite differences, Macmillan,
London,
1933.
4. N. E. Norlund, Differenzenrechnung,
Honeywell,
Minneapolis,
Springer, Berlin, 1924.
Minnesota1
1 The author is no longer employed at Honeywell. Reprint requests should be
addressed to the author's home, 1650 Vincent Avenue North, Minneapolis, Minne-
sota 55411.
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