This research was supported in part by National Institutes of
Health, through Grant No. 6-822-E501633-01.
SUMMATION METHODS, I
by
R. C. Grimson
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 989
March 1975
SUMMATION METHODS, 1*
R. C. Grimson
A.
Introduction.
The objective of thi s work is to present new and
little known series operations that are particularly applicable in
obtaining multiple generating functions for combinatorial and
probability problems.
Infinite series appearing herein are to be
regarded as formal power series or else appropriate assumptions
regarding convergence must be made; no further reference to convergence will appear.
We are concerned with methods of changing the bounds, the
order of summation and/or the summand of such series as LAf(i1, ... ,in)
(A is a set of conditions for the indices) so that the result is in
some 'desirable' form.
What constitutes a desirable form depends on
the problem; frequently, attempts are made at (a) interchanging the
order of sUlTIllation while keeping the summand unchanged, (b) making the
lower bounds constant (usually zero or one) or the upper bounds
00,
and
compensating by making appropriate changes in the argument of the
summand.
Usually (a) is used in achieving (b) and vice versa.
Looking
at it another way we see that the emphasis is on delineating useful
arrangements of the terms of multiple series.
Part of the results are independent of the nature of the summand;
in other cases the results depend on general properties of the summand;
e.g., it is symmetric,, the coefficients are cyclic.
For the most part
we consider cases where the only restriction on an index is that it
*Supported in part by NIH Grant No. 6-822-E501633-01.
-2-
ranges consecutively from one integer to another (or to 00).
However, many
cases occur where the conditions on the indices are more complicated.
As
alluded to earl ier, there often occurs a ki nd of reci proci ty between a
sum having a complicated summand with simple conditions on the indices
and that of a sum having a less complicated summand but a more involved
system of conditions for the indices.
For example, we often have identities
such as LA1= L~j=of{i,j) where A is a set of conditions on i and j (more
•
complicated than the conditions on the right, 0 ~ i, j
~
n) and where
f{i,j) is a function (more complicated than the summand on the left).
In order to maintain reasonable brevity, we shall not consider i
familiar properties of the usual moment, enumerative, etc. generating
functions; our work is more general.
Nor shall we exhibit the many
summation methods found in books on finite differences (Poisson's formula,
method of inverse differences, etc.) and the many interesting formulas
found in a variety of works including Gould [13J, Knuth [32J, Rainville [35J,
Riordan [38J, Schwatt [41J, Jolley [31J, and Mangulis [33J.
The stress
here is of a different, though complementary, nature.
A few words about the arrangement of the material is in order.
Bear in mind that this work represents a collection of formulas of certain
types.
The formulas are contained in Part B; there are 67 of them (not
counting ramifications).
Some of the latter formulas include some intro-
duction or discussion and are therefore lengthy.
We continue to regard
these more lengthy statements as formulas (rather than sections, etc.)
in order to maintain continuity of the format; the earlier formulas are
shorter and would not be considered sections .. Some formulas contain several
equatio,ns.
These equations are referred to by small letters (numbers
,
.
identify formulas). If an equation of an earlier formula is referred to,
the formula number followed by the equation letter is given, e.g., (32.C).
-3-
No claim to completeness is made.
in such a work as this.
Clearly, no claim can be made
In fact, several aspects suggest important avenues
for further research.
A note regarding any errors found here will be appreciated by
the writer.
This writer hopes that the reader will find at least a few new
results that are interesting and/or applicable.
Before listing the formulas, we introduce the following
notation:
Meaning of Symbol
Symbol
If found next to a formula, it means that n
*(n)
may be replaced by
00
in that formula.
[x]
The largest integer
{xl
The smallest integer 2: x.
I if i=j
Kronecker data = { a if i#j
o..
1 ,J
~
x.
A primitive mth root of unity (am#l).
Other notation is given in context.
Some conditions are understood.
nonsensical if n < a.
For example, the sum L~=a is
Care should be taken in applications of more
complicated sums to see that all conditions are satisfied.
B.
Formul as.
n
1.
L
j=a
f(j) =
n
L
j=a
f(n+a-j) .
-4-
n
2.
3.
l
j=a
n
L
j=a
n
4.
5.
L
j=O
n
L
j=a
l
j=a+b
r
L
f(j) =
, a 2. r
f(j)
[(n-l)/2J
j=O
j=O
[n[2J
'f..
j=[(a+1)/2J
.
L (-1) J f (j) =
n
L
j=r+1
L f(j) +
f (j) =
.
L (-1) J
n
[nl2J
'f.
'f.
j=[a/2J
n, *(n).·
f (2j +1), *(n) .
[(n-1)/2J
L
j=O
b-1
[(n+k)/bJ
k=O
j=1
l (-1) k
<
f(n-j).
[(n-11/2J
f (2j) -
j=O
f (j) =
j=1
L
f (2j) +
[n[2J
j=O
7.
*(n) .
f(j-b)
f(j) +
j=a
f(j) =
n
6•
n+b
f(j) =
L
f (2j +1) ~ *(n) .
..
f (bj - k)
(-1) bJ
(See [42J; there are a number of interesting similar formulas in
this work.
nb
8.
I
j=1
b-1
L
f(j) =
f (j)
3j~1=r(mod
j~
10 .
2)
\'
1 <m
f(kn+j) .
= l
j odd
f (j)
=
[(m+1)/2J
L
j=1
j<:m
L
3j~n(mod
L
k=O j=1
l
9•
n
2)
.
f (j) =
\'
[mL2J
f (j) = . 2' f (2j )
j odd
j=1·
L
jSn
.
(2j -1 ),
r even
-5-
11.
[n/2]
/.
f(j) = I
f(j) = I f(2j)
3j-1=r{mod 2)
j even
j=l
j<m
j<m
r odd
12.
13.
/.
f(j) .= I f(j) =
3j=r{mod 2)
j odd
j<m
j<m
k '
00
I
j=O
ak+J'm x +Jm = m-
1 m
k'
,I
J=l
a~- J
[(m+1)/2]
I
j=l
00
I
t=O
f(2j-l)
't
at(a~ x)
(This is the multisection of series; am is an i th root of
unity and am 1 1.)
n
2
n
n
14.
(I
15.
n
j
n
2
n
2 I
I f(j) f(k) = (I f(j)) + I f(j)2
j=O k=O
j=O
j=O
16.
j=l
18.
j+ak=n
f(j,k) =
j
I I
j=l k=l
I
k=O
f(i)2 g(k)2 -
I
l<k<j<n
(f(k) g(j) - f(j) g(k))2 •
.
f(n-ak,k)
n
I
I
n
j(j-l)
f(j,k)=
j=a k=b
I
=
en/a]
I
n
17.
f(j) g(j))
/.
j=a k=a{a-l)
I
n
I
k=b j=max(a,k)
f(j,k) =
f(j,k), *(n).
n(n-1)
n
/.
. I.
f(j,k) ,
k=a{a-1) j=1+[{1+14J(:j)/2]
*(n)
-6-
19.
n
rj
r
2 f(j,k)
j=a k=a
n
20.
I
I
I
ra-1 n
I
I
r
nr
n
I
L
L
j=a k=a
I
L
j=a k=b
+
n
I
k=ra j=1+[(k-1)/r]
f(j,k) ,
1.
*(n).
minfk,n)
L
[n/r]
f(j,k),
n
I
I
k=a j=rk
[aIr] tr-1
I
f(j,k) ~
rn
I
f(j,k),
k=O j=[(k+r-1)/r]
f(j,k) =
n [j/r]
23.
Y
I
n [j/r]
22.
~
1, a
k=O j={k/r}
rn
f(j,k) =
j=O k=j
~
*(n).
rj
I
f(j,k) +
k=a j=a
f(j,k) =
j=O k=O
n
21.
rj
=
L
f(j,k),
*(n)
*(n) .
f(j,k)
k=b j=a
t+s-1
r(t+s)-l
I
L
f(j,k)
k=b j=tr+rok,t+1+2rok,t+2+.··+(s-1)rok,s_1
t+s
+
L .
n
Y
.. f(j,k) ,
k=b j =rl t+2)
where t and s are determined by t = [aIr] + 1 and
n-tr= sr + p, (0
n
24.
j
L I
j=a k=b
klj
n
f(j,k) =
I
~
p < sk).
[n/k]
l
k=b j={a/k}
f(jk,k),
*(n).
*(n)
.
-725.
~
J.
n [n/kJ
L f(~ ,k} = L Y f(j,k),
k=b j=la/k}
j=a k=b
l
*(n).
klj
00
26.
00
L
L
j=a k=j+a
n
27.
29.
30.
r
L f(j,k) =
n
00
00
I
L
k=r-n j=r-k
n
l
a
j=l k=l
*(n) .
n
L f(k+n-j, n-j+a) .
00
k-a
L L
f(j,k) =
f(j,k) .
k=2a j=a
a
[\,!JJ.
l
f(j,k),
k=a j=k
j=a k=j+a
~
n
I
L
L I f(j,k) = I
j=a k=j
f(j,k).
k=2a j=a
j=O k=r-j
n
28.
L L
f(j,k) =
r
L
k-a
00
n
_[InJ
L La
f(J,k) -
f(jk)
k=l k=j
(See C12J .)
2n_1 n
n 2i -1
31.
L L
j=l k=l
f(j,k) =
L
L
j=l k=1+[loQ2 jJ
f(j,k).
(See C12J.)
00
32.
00
L I
j=O k=O
00
f(j,k) =
I
n
I
j=O k=O
f(j-k, k).
-800
33.
L L
j=O k=O
00
34.
00
00
f(j,k) =
j
L
j=O k=O
00
L L f(j,k) = I
j=O k=O
Cj/2J
Cj/2J
I
j=O k=O
00
00
f(j-2k, k) .
f(j-k, k) .
00
00
=.
L. f(jl+j2+ .. ·+j r' j2+j 3+· .. +j r' ... , jr-l+j~, jr)·
J1,···,Jr=o
-9-
j2
L
f(j1' j2' ... , jn)'
*(r) .
j =0
1
00
39.
00
+ ••. + f (j l' j 2' ... , j n-1' j n + k + 1, k + j 1 + ... + j n-1 + I)}
.
-10-
min(jl' j 2'''' ,jn)
00
40.
L
L
jl ,j2"" ,jn=O
00
f(k, j1' j2"'"
00
r
=
k=O
ooL
.
f(k,j1,j2, .. ·,jn)·
J 1 ,J2, ... ,J n=k
k=O
00
max(i,j)
00
I
L
42.
i,j=O
f(i,j,k)
k=O
00
L
=
{f(i+k, i+j+k, k)+ f(i, i+j+k+1, i+k+1)
i,j,k=O
+ f(i+j+k, j, k) + f(i+j+k+1, j+k+l, i+k+l}}
00
L
f (i +k, i +k, k) .
i,k=O
(There are other similar expressions for
'\~
L.,
,\max(i ,j) f(i JO k).)
,J=O L.k=O
' ,
0
jn)
-11-
k
00
=
I (~).
i=l
44.
00
I.
s (j l' ... j i' 0, ... ,0) + S(0 ... ,0).
Jl' .. .,Ji=l
'
Let s(j1' j 2, ... ,jk) be symmetric in jl,j2, ... ,jk and let
I *s(jl,···,jk)
be the symmetric sum over jl, ... ,jk, ... ,jn
(the sum of S(jl,.··,jk) over all k-combinations of jl'·· .,jn).
45.
Let
I*
Then
be the symmetric sum over s out of k characters.
If f is symmetric, then
I*
=
where the symmetric sum on the left is overs out of k
XIS
and the
symmetric sum on the right is over s out of k n's and does not
extend beyond the braces.
-12-
46.
Suppose that n ~ 1 and aI' ... , an are positive integers that are
r~latively
prime in pairs.
Let A = a1a2 ... an'
Let
f 1(x), ... , fn(x) be functions of x of period 1 such that
there exists the relation L~:6 fi(X + r/k) = ci k) fi(kx) where
ci k) is independent of x.
Then
and, more generally,
This is due to Carlitz [4J and [5J where there appear some
applications.
47.
If 9 is a continuous increasing function and if g(i) is integral
for i
E
{a-1,a, ... ,n} and if g- denotes the inverse function of
9 then,
ng(i)
(a)
L
I
i=a j=g(a)
g(n)
f(i,j) =
L
L
j=g(a) i=!+[g-(j-1)J
f(i,j),
*(n)
This formula is useful in some problems requiring the interchanging
of the order of summation.
others are familiar with it.
H. W. Gould (see [14J) and probably
A proof, one which we adopt here
-13because of its revealing applicable structure, is given by Towe [43J.
Proof of (a).
The left side may be expressed as the sum of the
entries of the following table where x denotes
f(i~~}
evaluated for the
indicated values of i and j:
j
i
g(a) g(a}+l g(a}+2 ... g(a)+n O g(a+1) g(a+1)+1 ... g(a+1}+n 1 g(a+2} ... g(n)
a
x
a+1
x
x
x
x
x
a+2
x
x
x
x
x
x
x
x
n
x
x
x ...
x
x
x
x
x
x
Note that
g(a+k}+n k+1=g(a+k+1).
(b)
Reversing the order of summation, the sum may be expressed as
g(n)
(c)
L
n
L
j=g(a) i=b(j)
f(i,j}
where
(d)
b(j)=a if j=g(a)
and
(e)
b(j)=a+k+1 if
g(a+k}+1<j~g(a+k}+nk+1,
k=O,1,2, ... ,n-a-1.
Then, in order to obtain the right side of (c) we need to show
that b(j)=l+[g- (j-1)J.
We do so by considering (d) and (e) separately.
-14-
From (e) we must show that if
a+k=[g- (j-1)J.
an integer.
tradiction.
If a+k>[g- (j-1)J, then a+k>g- (j-1) because a+k is
Since 9 is increasing, g(a+k»j-l.
contradiction.
g(a+k+1)<j-l.
g(a+k)<j-1~g(a+k)+nk' then
Next, if a+k<[g- (j-1)J then
But this is a
a+k+1~g-
(j-1) so that
Then by (b), g(a+k)+n k+1<j-l; we have another conTherefore, a+k=[g- (j-1)J as required.
On the other hand, from (d) we consider the case j=g(a).
Since 9 is increasing and since g(a-1) and g(a) are integers
g(a-1)<g(a)-1<g(a).
Then a-l<g- (g(a)-l)<a because g- is increasing on [g(a-l), g(n)J.
From this we see at once that
a=l+[g- (g(a)-l)J.
Thus (a) is proved.
The formula, with only minor changes in the above proo'f, is
also valid if n is replaced by 00; this case frequently occurs.
Note that if b < g(a), then
n g(j)
n g(j)
n g(a)-l
(f) .l l f(j,k) = l
l f(j,k) + L l f(j,k).
J=a k=b
j=a k=g(a)
j=a k=b
Equation (a) now may be employed to evaluate the first sum on the
right side.
A sum of the form
n
(g)
l
g(j)
L
j=a k=h(j)
f(j,k)
is usually difficult to work with.
One approach is to put
-15-
r(j) = g(j) - h(j) and e(j,k) = f(j,k+h(j)) so that it becomes
r(j)
n
L
(h)
L
j=a k=O
e(j,k)
.
Now, perhaps (f) can be applied; however, e(j,k) may be unwieldy.
An alternate approach may be possi bl e;
~
gij)
I
L
f(j,k) -
j=a k=h(a)
n
n
L
h(j)-1
L
j=a+1 k=h(a)
(g) is
f(j,k)
n
h(j)
n
f(j,k) - ( I
I
f(j,k) - I f(j, h(j)))
j=a k=h(a)
j=a+1 k=h(a)
j=a+1
= L
g(j)
L
n g(j)
n hij)
n
I
f(j,k) - I
I
f(j,k) + f(a,h(a)) + L f(j,h(j).
= I
j=a k=h(a)
j=a k=h(a)
j=a+1
Perhaps (a) and (f) can be applied to some of these sums.
For a simple example, suppose we are interested in the sum
_n
j2
.
S-Lj=O Ik=j f(J,k).
Then, as in going from (g) to (h),
s=Ij=o I~~g-l) f(j,k+j) and the order of summation may be inverted
according to formula 18.
Under the same conditions of function (9) above, we note
the following [12J:
n g(i)
If g*(n) = card {mlf(m) < n} then
g(n)
n
f(i,j) = I
1.
f(i,j).
1=1 j=l
j=l i=g*{j )+1
L I
-16-
Formula (a) is sometimes useful in interchanging the order of
summation of multiple sums, provided the conditions on 9 are met.
Suppose g(i,j) is continuous and increasing in both variables.
Then by two applications of (a) we obtain
n
g(i,j)
I
g(n,n)
1.
i,j=a h=g{a,a)
where
g~
f(i,j,k) =
1.
h=g{a,a)
n
n
I
I
i=1+[g~(h-1,n)J j=1+[g~(i,h-1)J
denotes the inverse of g- with respect to h.
f(i,j,k)
Other similar
formulas may be obtained for multiple sums.
48.
Let y denote an increasing continuous function on the real numbers
such that y(n) is a positive integer if n is a positive integer and
y(O) =
o.
Also let y- be the inverse function of y.
Then
00
(See [18Jand [25J.)
49.
Under the conditions of formula 48,
00
00
50.
Under the conditions of formula 48,
[y - (mi n (j 1' ... , j n)J
I
l:
f(k,jl'···,jn) =
ji'··· ,jn=O
k=O
00
00
00
-17-
51.
Under the conditions of formula 48,
ro
ro
L
'i _
.
.
f(k,jl, .. ·,jn)
jl, ... ,jn=O k=h (max(J!'···,J n)}
I
I
ro
=
L
.L
f(k,jl,· .. ,jn)
k=O jl, .. ·,jo::y(k)
\
I
52.
The q-Eulerian function Hk (xlql, ... ,qk) may be defined symbolically by x-I rr~=I(1 - qjH) if k ~ 1; in addition, HO = 1. Roselle
[39; (3.2) and (3.9)] proved the following:
For further properties of H, see [3], [8] and [39].
We let y denote an increasing continuous function on the real
numbers such that y(O) = 0 and y(n) is a positive integer if n is a
positive integer.
Also, we let y- be the inverse function of y and recall
that {x} is the least integer
~
x.
Then
(c)
-1 .. qk-1 H (zlql
- 1 , .. .,qk-1) .L\? c ·(ql·· ·qk z )y(j)
= ( l-z )-1( -1 )k ql'
k
J=O J
Proof is as follows:
-18-
[y-(min(n 1,···,n k))J
n1
nk t
):
cJ' q1 ... qk Z
J C=O
[y-(min(n 1,· .. ,n k))J n
n
=
L
/.
.~
Cjq11 ... qkk zt
n1,···,nk+O t~max~n1, ... ,nk)
J-O
co
-1
=(l-z)
co
L
_
[y-(min(n 1,···,n k))J
L._
n1, ... ,n k-O
J-O
(formula 41)
n1
nk max(n 1,···,n k)
cj q1 .. ·qk z
(formula 41)
[ J)
(property of y)
(equati on (a))
co
-1 H (z Iq1
-1 , ... ,qk
-1 ) .l~ c ' (q1
= ( 1-z )-1( -1 )k q1-1 ... qk
k
J=O J
(equati on (b))
Application 1.
Let f(n) be a nondecreasing sequence of positive integers
and define the distribution function, D(f), by
D(f (n)) = card {k If (k)
~
n; k = 1, 2, ... }.
Note that
(d)
Y
n=l
D(f(n)) xn =
Y
n=l
L
f(k)~n
xn =
Y Y
k=l n=O
xn+f(k) = (l-x)-l
Y
k=l
xf(k) .
.
-19-
Therefore, if we-put Co = 0 and Cj = 1 in (c) and then use (d),
we have
(I -1 ,· .. ,qk-1)
qk-1 Hkzq
00
\' (ql .. · qk z )y(j)
j=l
L
This gives an interesting q-generating formula for distribution functions:
00
.L
J=l
•
D(y(j))(ql··· qkz )J
Application 2.
In (c) put o(j) = jm, Co = 0, cj = l(j>l). Then
(e)
.m
00
= (1-z) -1()
- 1 I (ql'"
-1 k ql-1 ... qk-1 Hk(I
z ql-1 ' ... , qk)
j=l
qk
In the case that k=l, (e) may be compared with some results in [12J,
[44J and other similar formulas of number theoretic interest.
z) J
-20-
53.
(a)
Under the same conditions of the summation formula (52.c), we have
'"
l
t
L
t=O n1, ... ,nk=O
-1 ·qk
1-1 H (I
-1
-1
= ( l-z) -1 {-1 }k q1··
k z q1 , ... ,qk )
Thi s formula may be compared with the partiti,on fonnul as of [18].
of (a) is as follows.
Proof
The left side becomes
= (l_z)-l '"l c·
z=o J
The last step is obtained by applying (52.a).
Using (52.b) this becomes
the right side of (a); this completes the proof.
Application.
"y(x) = xm, then
If, in (a), we put
Co
= 0, cj = 1 (j
~
I) and
-21(b)
-1 ( )k -1
-1 (I -1
q-1) ~ (
){J,l/m}
= (1-z )
-1 q1 ... qk Hk z q1 ... k j~l Q1··· qk z
The Eulerian polynomial, ak(x), may be expressed as (see [18, p. 303J):
Also, x(x - l)k Hk(x) = ak(x) where Hk(x) is the ordinary Eulerian
function Hk(xll,l, ... , 1).
relation
Hence, from (b) we have the interesting
(c)
If we put q1 = Q2
= ... = qk = 1, then (c) becomes a new product
formula for Eulerian functions (and hence polynomials):
(d)
1-z ~L
~L
(.mln
, ( n ,·.·,n ))m z.
t
Hk(z) Hm(z) = (-1) k+m --z-1
k
t=o n1, ... ,n k=0 .
Notice the curious symmetry in m and k.
Dividing (d) by (_l)k+l (l - z)/z, expressing the left side
as a power series in z and comparing coefficients we get
-22-
where the An,j are the Eulerian numbers defined by (x - l)n Hn(x)
~n
j-l .( n > 1 ) . Evidently (e) is a new interesting relation
-_ £.-1
J- An,J. x
for the Eulerian numbers. Special cases of (e) include Worpitzky·s
well known result xk = ,k
£s=l Ak,s (x+s-1)
k .
Expressing the right side of (e) as
L~=O (k~+h) Li+j=t-h Ak,i Am,j and then inverting [38; p. 106J we
find a convolution formula for Eulerian numbers:
(f)
L
i+j=t
A
k,i
A
')n,j
t
t-h-1
= £~ (_l)h (k+mh+l)
(mln
. (. n1, ... ,nk ))m .
£,
h=O
n1 , ... ,n k=0
Combinatorial significance of Eulerian numbers is well known
(see [37; p. 214J and [8J): An,kis the number of permutations of
{I, 2, ... , n} with k rises where it is agreed that a rise appears
on the left.
Thus there is a direct interpretation of (f); it gives
the number of permutations of {I, 2, ... , k} and {I, 2, ... , m}
for which there are a total of exactly t rises.
54.
The results of (52) and (53) can easily be generalized in some
di recti ons . Cons i der the fo 11 owing.
Let
y
be a functi on from a p
dimensional (real) space to the real numbers.
function of
y
as follows:
Define the distribution
-23Then,
A q-generating function for the distribution function is found by
conveniently defining c.
c.
.
=
'l"""P
Special note.
Thus, we may put ci1, ... ,i p
= 1 or
r0 if at least one of the i's is zero
"
L 1 otherwise.
In much of the remainder of this work we discuss some
general methods for determining power series which possess one or both
the following properties:
(a) the sequence of coefficients are periodic,
(b) the terms of the sequence of coefficients, when arranged in their
natural order according to the expansion of the series, change only at
specified positions. For instance, we may be interested in the series
a(O)x O + a(O)x 1 + a(O)x 2 + a(l)x 3 + a(l)x 4 + a(2)x 5 + a(2)x 6 + a(2)x 7
+ a(3)8 + a(3)x 9 + a(4)x 10 + ... where, further a(k) is periodic in I
k of period r.
Or we may be interested in this series with xk replaced
by xk/k or by some oth~r indicator. We point out that property (b) can
usually be effected by an appropriate composite of a function with the
greatest integer function; this idea was examined by Schwatt [4]J and [42J.
-24.,.
In some areas, our work is an extension of Schwatt's.
Also see the
bi b1i ography.
n
55.
L f(j ,[~])
j=O
[n/rJr-l
r[n/rJ+r-n-2
k=O j=O
j=O
= L L. f(j+rk,k) ""
00
56.
57.
58.
•
L f(j,
00
I
[~J) =
j=O
.
[n/rJ
j=O
r
k=O
00
•
00
f(j+n+l,[~J), r > l.
-
r-l
I
k=O j=O
n
L
f(j+rk,k),
r
> 1.
L f ([J..J) = r L f(k) - (r[n/rJ + r-n-l) f([IJ), r
L f ([J..J) = r L f (k) ,
j=O
r
a-I b-l
= L I
r > 1 .
k=O
00
Y.
f=0 s=o k=max\O,[(r-s)/bJ)
f(bk+s, (bk+s-r)/a,k)
if f(x,y,z) = 0 whenever y
a-I b-l
= L L
L
r=O s=O k=max(O,[(s-r)/aJ)
~
integer)
f(ak+r,k,(ak+r-s)/b)
if f(x,y,q) = 0 whenever z
(See El7J.)
> 1 .
~
integer)
•
-2560.
Suppose f k > 0, k = 0, 1, 2, .... A series in which the first p
terms are positive and the next q terms are negative, and in which this
pattern of alternating signs by groups is continued throughout, may be
represented by
[ 9+ kJ+[ k J
~
(-1)
L
p+q
p+q f
k
k=O
00
=
L
j=O
jm+p-1
L
k=jm
jm+m-1
L fk
j=O k=jm+p .
00
f
k
-
L
(m = p+q).
(See [21J, [24J and [42J.)
61.
A formula for the series that results from L~=l f(k) if the first
a terms are retained, the following b terms are removed,
the next a
terms are retained and the following b terms are removed, etc., is
given by Lk>l f(k+b[(k-1)/aJ).
00
g([j/rJ)
j=O
i=O·
L
62.
L
(See [42; p. 189J.)
~ gik) (k+~)r-1
L
f(j,i)
f(j,i) = L I
k=O i=O
j=rk
(Furthermore, the methods of formula 47 may be applied in order to
interchange the order of summation of the sums of k and i on the right
side.)
63.
Consider the sum
s =
~
f(qx + py)
where k extended over the lattice points
-26{(x,y) IO<x<p, O<y<q, qx + py<pq}, where p and q are relatively prime
positive integers and where f is a polynomial.
It is shown by
L. J. Mordell C34J that s is determined by the two formulas
q-1 p-1
L L
y=l x=l
pg-1
2
1=1
64.
f(qx+py) =
f(l) -
p-l
L
L f(gx+py)
k
f(lq) -
1=1
q-l
L
1=1
+
L f(2pq-qx-py),
k
L f(qx+pq)
f(lp) =
k
+
L f(pq-qx-py).
k
We present this formula because it falls into an interesting class
of sums, sums of the general form Lkf(Ck/mJ,Ck/nJ) where m and n are
Recall that am is a primitive root of unity
positive integers.
(amll) .
If a(k,i,j:r) is a function in i,
j~
and k and if it is periodic
in jwith period r then
00
\' a(k ' CkJ
ct
J : r) xk
m
'
n
k=O
L
r-l
=
00
L L
(j+hr+l)n-l
L
j=O h=O k=(j+hr)n
k
k
a(k,CmJ, j:r) x (periodicity: Formula 66a)
r-l
n-l
(")
.(
)
= L L L a(k+(j+hr)n,C k+ ~+hr nJ , j :r) xk+ j+hr n
00
j=O h=O k=O
r-1 n-l
00
= L L {L a(k+jn+hrn, Ck+jn+hrnJ , j:r) xk+jn+hrn}
j=O k=O k=O
r-l n-1
m
rn
= L L (rnf1 L arn-(k+jn)h L a(t,ctJ, j:r)(lX~n x)t (multisecti'on)
j=O k=O
k=l rn
00
t=O
m
-27-
f.-I n-l
==
L L
j=O k=O
= (rn)-l
= (rn)-l
(rn,-l
rn
.)
m-l
n- (k+Jn
or
h I I a(l+mt, t,j :r)(a h x) l+mt (Formula 56)
h=l rn
t=O 1=0
rn
00
L
r-l n-1 m-l rn
(.)
h}
a~n- k+Jn h {I a(l+mt,t,j:r)(a x)l+mt
j=O k=O 1=0 h=l n
t=O
rn
L I °L
r-l n-l
00
L
00
rn
0
L a(s,t,j:r)arn-kh-jnh+hs xs
L L L L
j=O k=O s=O h=l l+mt=s
l<m-l
rn
(Compare with formula 59.)
65.
If a(i,j:r) is a function in i and j and periodic in i with
period r then
I a(k,[~J:r)xk
= 1 rIl
k=O
n
r 1,j=0
66.
00
L
k=O
=P
a~-j(l+l)+l(k+nt)
a(j,t:r)x nt .
If c{x:r) is a function of x of period r then
cn 9+kJ +[-LJ:r) f k
p+q
p+q
Mk+p
00
=p
I
nIl
k=O t=o
L
c(2k:r)
00
(t+l)r-l
k=O
L
L
t=O k=tr
If~
j=Mk+!
+q
c(2k~r)
00
M(k+l)
k=O
j=M(k+l )-q+l
I c(2k+l:r)
Mk+p
L
j=Mk+l
L fj
00
f J. + q
I
t=O
(where M=p+q)
(t+l)r-l
L
k=tr
M{k+l)
c{2k+l:r)
L
t
j=M(k+l)-q+l
-28r-1
00
=p
L L
t=O k=O
r-l
M{k+tr)+p
c( 2(k+tr ) :r )
00
L
j=M{k+tr)+l
r-1
L L c{2(k+tr+1) :r )
t=O k=O
(Compare with formula 60.
67.
r-l
M{R+tr+l)
L
fj
j=M{k+tr+l)-q+l
M{k+tr+l)
= p L c{2k:r) L
L f +q L c{2k+l:r) L
L fj
k=O
t=O j=M{k+tr)+l k=O
t=O j=M{k+tr+1)-q+l
00
M{k+tr)+p
fj + q
00
M=p+q.)
Let a{i, j:r) be a function in i and j, periodic in j with period r.
If nand r are positive integers then
00
00
n-l
L a{k, .~ :r) = L L a{k + nj, j:r)
k=O
n
j=O k=O
n-l
00
= L L
k=O h=O
n-1
00
(h+l )r-l
L
j=hr
r-1
= L L L
k=O h=O j=O
r-1
00
= L L
j=O h=O
a{k + nj, j:r)
a{k + n{j + hr), j + hr:r)
{j+hr+l)n-l
L .
k= (j+hr)n
a{k,j :r).
{There is a direct extension to multiple sums of
Application.
Let i be {_1)1/2 and consider
-29-
If fk(x) = xk, then this series becomes (l_x n )(l_x)-I(I_x n exp(2TIi/r))-l,
The more interesting case is where fk(x) = xk/k!; so we consider
(b)
E(r,n; x) =
Ik=O expeTIir [kJ)xk/k!
n
,
The following definitions are useful:
h=O
L
k= (j+hr)n
k
x /k!
J:
.-sx S(n, r,
j;
00
S(n, r,
j;
x)
=
s(n, r,
j;
s)
=
L
(j+hr+l)n-l
x) dx.
We find that S(n, r, j; s) = srn-jn-n (sn - l)(snr - 1)-1 (s _ I)-I,
In order to invert the Laplace transformation we apply formula 21
of [1, p. 232J.
S(n, r,
This gives
j;
x)
where the function <I>a, b(s) are rational; with a little manipulation
they may be determined using the formulas of [IJ. Then
x
1 rp
(c) S(n, r, j; x) = e /r + rn k;2
w~-n-jn(l_w~)
1 _ wk
(
) (02.J'
exp wk x ,
< r)
where w2' w3' ... , wrn are primitive (rn)th roots of unity.
-30-
Applying (a) and (c) to (b) we get
E(r, n; x) =
r-l
L
j=O
exp
e
1T: j } S(n, r, j; x)
x
rnw 1- n- jn (1 - wn)
L exp (21T: j) ~ + 1- L k
k
j=O
r
rn k=2
1 - wk
r-l
=
.
1
wkl-n-jn(l-Wnk)
1 r~
(21T i J\ ~n
= e 01 r + rn .l exp
l
l-w
exp(wkx)
r J k=2
k
'J=O
x
where 01 ,r is the Kronecker delta. This equation has several interesting
consequences; for instance, it generalizes some results of [21; section 2J
concerni ng a1ternati ng exponenti a1 seri es.
Note the fo 11 owi ng speci a 1
cases.
E(l, 1; x) = eX
co
E(2, 1; x) =
L
k=O
co
E(2, n; x) =
L
k
(_I)k ~! = e- x
k
(_I)[k/nJ ~!
k=O
co
E(4, n; x) =
k
i[k/nJ x
k=O
x!
L
Also, note that E(r, 1; x) is the exponential series for roots of unity.
While the specific results are interesting, we emphasize the general
method used in obtaining the results.
This method may be applied to
other power series with cyclic coefficients.
-31-
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1.
Bateman, H.
1954.
2.
Berndt, B. C. and Schoenfeld, L. Periodic analogues of the ELilerMaclaurin and Poisson summation formulas with applications to
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3.
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4.
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5.
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6.
Carlitz, L. The generating function for max(n 1 , ... ,nk)'
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Carlitz, L.
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L.
13.
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14.
Gould, H. W. The interchange of order of summation in double sums.
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15.
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Grimson, R. C. Enumeration of symmetric arrays with different row
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17.
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18.
Grimson, R. C. Some partition generating functions. Proceedings of
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20.
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22.
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Grimson, R. C.
appear.
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26.
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27.
Grimson, R. C.
28.
Grimson, R. C. The evaluation of a sum of Jacobsthal, Norske,
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Grimson, R. C.
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30.
Grimson, R. C. The number of monomer-dimer configurations in
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32.
Knuth, D. E.
1969.
An exponential series and roots of unity.
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Fib.
To
To appear.
The generating function for {M r (n 1, ... ,n k)}m.
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Mangulis, V. Handbook of series for scientists and engineers.
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Rainville, E. D.
Special functions.
..Uber die
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Kettenregel n-ter Ordnung, Math. Ann.
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..
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J. Undergraduate