cos*> +
1
-
2jBw c o s * > +
1
+
and
(3.15),
Rco ' 1 +
+ 1)
fi sing(X
flV
1
=
1 -
sin cos ? +
iJV'
(3.6),
y
Using Thus w =
RW
we may conclude that
x
y
W
1
x
(3.14)
i?V
Rco s i n
1
x
N
1 R
— ffdJ COS
x
(3.14)
1 -
sin
2flco cos
+
#V*
and ( 3 . 2 ) , these formulas m a y be written in terms of n(X) = E C ( X , j)A'/(0),
y-o
+1
BnW
= E C(\ - 1 - i, n)A" /(j).
y-o
Observe that the remainder differs from the integral form of the remainder of Taylor's formula in exactly the fashion that (5.1) differs from (5.2). This must be the case since Taylor's formula is an identity due to repeated integration by parts whereas Newton's formula is an identity due to repeated summation by parts. Observe that (6.3) is valid with no regularity assumptions. If A f(j) is independent of/, then n+1
R (\)
= A
n
n + 1
/ ( 0 ) E C(A - 1 - j , n),
y-o
and, using (5.3), R (\) n
n+1
= A /(0)C7(X,n + l ) .
Hence, for this case,
/(X) = y-o EC(X,j)A'/(0). We return to the general case. Note that #»(X) = 0 for X = 0, 1, • • , n. Let V
/ X
°'
( X
'
n ) = =
C(X - 1 -j,n) C(X,n + l) '
X
>
"
Then aj(\,n)
(6.4)
^ 0,
j = 0, 1, . . . ,X -
1;
2^ay(X, n) = 1. y-o
We can write the remainder as (6.5)
R {\) n
= C(X, n +
1) E
«0 =
1.
The coefficients can be real or complex. I f / ( 0 ) , / ( l ) , • • • , / ( n — 1) are specified, the solution is uniquely determined for all X. Equation (7.1) might better be called a recurrence relation; the term difference equation is used for historical reasons. The polynomial E?-o a ~jt is called the characteristic polynomial and the equation J
n
(7.2)
£ a _/ « 0 n
is called the characteristic or indicial equation. The roots of the equation
E «/ = 0
(7.3)
are the reciprocals of the roots of (7.2). The sequence g is sometimes called the forcing sequence. We begin by considering the first-order equation (E + aOf = g.
(7.4) From (4.4),
=
-i/(0). CO
CO
Hence L
±
^
=
/
CO
I/( ) 0
+
j .
CO
Therefore, (7.5)
/(0)
/ = 1
+
+ y^T^— g.
a i co
1
+
a i co
From (3.5) and (3.6),
(7.6)
X
X_1
/(X) = / ( 0 ) ( - a ) + 2 ( - « i ) - V O ) . 1
This analysis exhibits a number of features which hold true in the nth-order case. The particular solution is expressed as a convolution. As we shall see below,
WIDE
GAL
P-55
WIDE GAL
P-55
Gal. 7 MT 3170 p. 30 Take l-14-10gx 18—l-f>-65 62 the form of g may be such that we can evaluate this convolution without writing it as a sum. It is not necessary to obtain the general solution of the homogeneous equation plus a particular solution of the inhomogeneous equation and then to fit initial conditions. This is all done in one step. The dependence of the solution on the initial conditio!^ is explicitly obtained. We turn to the second-order equation
+ a )f
(E* + aiE
= g.
2
Using
we find that (
7
7
,
)
=
/(0)(1
+ a m ) + /(l)o>
coV
+
Let the zeros of the characteristic polynomial + a 2 be pi and p-2 with Pi ?^ pi. The partial fraction expansion (see Appendix A) of (7.7) leads to /(x)
=
—
l
—
pi —
G(X) = / ( 0 ) [ P 2 ( 7
'
8 )
+
[GM
s(x)j,
Pi
= /(O)lfli
X+1
- pi
X+1
+
- pi )l + / ( 1 ) U P. P2 J + / ( D t p / - Pi ], ai(P2
X
X
-
Pi")
X
Pi' -
X
s(x) = E ( / ( i ) ( ^ " - p ' w
x _ w
)-
Let the characteristic polynomial have a double root at 1 =
(1
X
-
Pi « )
(
P l
. Using
+ 1) \
X
P 1
2
it is easy to show that
2
/(X) = a M / W M I - X)] +/(1)X} +
(X - 1
-
jV-*-'g(j).
The same result may be obtained from (7.8) by calculating the limit as P2 —> p i . Observe that pi and P2 are, in general, complex. If the coefficients and the starting values are real, the solution is real for all X. In the form of the solution given by (7.8) it is not clear that / is real. We give an example below to show that all calculations may be done over the reals. We turn to the nth-order equation n
E y-o ctn-jE'f = g,
«0
Using (4.9), y-i j
E'f-
r f-
Er /(Jk), H
fc-0 y—o
y—o
it—o
Interchanging the order of summation and setting r = 1/w, we find
E y-o
ccjJf
= E «,*/(*)e « fc-o
y
. y-o
Let (7.9)
P («) = E r
a^.
y-o
Then the general solution may be written simply as
+
« v
Then the general solution may be written simply as CO
^ (7.10)
, J
£
k
Ar-0 =
, ''
DTA PN(W)
CO 0 P (co)' n
PQ
Assume that the roots of the characteristic polynomial are distinct. Let
O
(7.11)
C (O) = R
Eor-V.
A calculation shows that Ai
C/PN-X-^CO) _ £ P„(CO)
i - 1 1 — pi CO '
Ai
Cn-l-k(pi)
= -p,
C„'(p.)
Furthermore, =
E 2-N c„'( ,r P
Therefore, /(x) = E
(7.12)
f(k) t
" - " } » y + g
c
E
£
1
_
t
or (7.I3)
/(x) = E
x
i-1
w
E
C (P.; *-0
t w * ( w ) / ( * ) + E »(*) E Jfc-0
n
i-1 C
*
n
(pt)
It follows from ( 7 . 1 1 ) that y*J> pi
1
* ft
/
(7.14)
-CUiM = coC(co) + r
ttr+1
.
Hono% fep ^iixed, all the C«_i_*(p») may be obtained by synthetic division wither , ^JMeA-vtn multiplications. Using the C -i-k(Pi), C '(pi) may also be obtained by i^ 7 synthetic division. (For material on synthetic division, the reader is referred to
f
n
*
n
Kunz [15, pp. 1 9 - 2 1 ] . )
We turn to some examples. The first example shows how all calculations may be done over the reals. Given (£
2
+ 2 £ + 4)/ =
^
solution is _ / ( 0 ) ( l + 2CO) + /(l)CO ;
2
1 + 2CO + 4CO
Writing l + 2W = l + W + W and using ( 3 . 1 4 ) and ( 3 . 1 5 ) , or using ( 3 . 1 8 ) and ( 3 . 1 9 ) , we obtain /(X)
= 2 [/(0) X
(cos | TTX + ^ - sin | t t x ) + / ( l ) ^
sin | t t x ] .
To illustrate the importance of distinguishing between constant sequences and scalars, we consider (E
2
-
3E + 2 ) / = 1,
/ ( 0 ) = 0,
/ ( l ) = 0.
The right side of this equation is the constant sequence consisting of ones. The solution is 2
1 -
Since
1 2
3CO + 2CO 1 -
2 CO
(1 -
2CO)(L - oof'
WIDE GAL P—60
WIDE GAL P—60
Gal. 8 AIT 3170 p. 34 Take l-14-10gx 18—1-5-65
02
1 2
1 2
(1 - 2co)(l - co)
1 - 2co (1 - co) ' X
2-
/(X) = As our last example, we consider (E - 1 ) / = X!,
X - 1.
/(0) = 0,
2
/(5) = 1.
Then 1
1 — CO 2
CO
and /(A)
=
i/(D(l -
X-1 X
("1) ] +
-
[1
(-D^Jj!.
y-o
Since / ( 5 ) = 1 = / ( l ) + 7, /(X) = - 3 [ 1 - ( - 1 ) * ]
+
[l -
i £
( - u ^ - t y .
y-o 8. I n v e r s e O p e r a t o r s of t h e D i s c r e t e C a l c u l u s :
-1
A ,
M~K
-1
We define A / to be
the solution of the first order equation A*> = /. 1
Clearly, A" / is determined only up to an arbitrary constant. Despite the symbolism in common usage, A does not commute with A. It plays only a secondary role in the theory of generalized sequences just as indefinite integration plays only a secondary role in Mikusinski's theory. From (7.5), -1
1
(8.1)
A" / = *LM + / = riTftO) + co,/, 1 — CO 1 — CO where A /(0) is just an arbitrary constant. Hence -1
X
(8.2)
A-Y(X) = (1) A-V(0) + E / C 0 , x
where ( l ) serves to remind us that this constant is added in for each X. We define M~ / as the solution of 2
Mip = /.
Hence 1
(8.3)
AT / =
+j%-fI ~T CO
1
M-ftO)
+
2urf,
CO
or, M~ f l
= (-i) M-y(o)
+ 2E
x
(-D^-yo-).
y-o l
l
Thus M~ is related to the operator v. Observe that whereas M and M~ are not inverses, v and n are inverses. For material on A and M~ the reader is referred to Jordan [13, Chapter III]. As an example, we calculate -1
l
y
-i„x
C
, 2co
1
1-fco l - | - c o l — aco = C\-l) + _ ? _ a \ x
a + 1
9. H e a v i s i d e a n d D i r a c S e q u e n c e s .
Let
9. Heaviside and Dirac Sequences. Let = 1,
X ^ ],
(x,i) = 0 ,
x -*
w
(1 -
CO)* 1 -
CO = CO
co
(1 -
+I
CO)* '
A ( x , y ) = c/~*c(x, &).
Therefore
7 ( x , i ) relations = r pbetween c{\k). C(X, k) and C(A X) with In a similar manner we can^ obtain k fixed. From k
co*
(1 -
we obtain
CO)*+*
(1 -
k
CO* )2A+1 (1 W
«) ,
WIDE GAL P—63 GALLEY
9
MT
WIDE GAL P—63
PAGE
3170
39
1/U/10B-GX
C(X, k) = « V
(9.8)
fr+1
62
x
( - l ) C ( i b , X).
Hence C(X, /c) is obtained by a (2k - f l)-fold summation of ( —l) C(fc,.X). As an example, let A; = 1. Note that x
X
(-1) C(1,X) = 1, - 1 , 0 , 0 , ••• . The sum of this sequence is 1, 0, 0, • • • . The sum of the new sequence is 1, 1, 1, • • • . Again taking the sum wc obtain 1, 2, 3, • • . Applying co to this and using (3.6), we obtain 0, 1, 2, • • • which is just C(X, 1), which is the left side of (9.8) for k = 1. From (9.8), T * C(\,k).
X
(-1) C(A;, X) =
k
P
+1
From the identity 2
+1
(1 +co)* = (1 + c o ) *
1 +1
CO* (1 + co)* '
we obtain C(k, X) = r y * ( - l ) C ( X , k). + l
X
Recall that \i is related to the mean operator. 10. D i f f e r e n c e E q u a t i o n s w i t h H e a v i s i d e a n d D i r a c S e q u e n c e s . Difference equations whose forcing term depends on Heaviside or Dirac sequences may now be easily handled. For a discussion of such equations using "classical" operator methods, the reader is referred to Tauber and Dean [21]. Consider the first-order difference equation O&'-0)/-7(X,j).
Then +1
J
1
1 -
J CO
1
£>..
' (1
0co
- co)(l
-
M '
and / ( x ) = /(0)/3 + x
-y(\j).
The dot in the last term of this equation is used to show that this is not a convolution. Consider the more general equation (E-p)f=
g(\)M\3).
Using (7.6), / ( X ) = / ( 0 ) ^ + E V p ( X - 1 - *). X
Jfc-0
The next two examples illustrate the treatment of an equation whose forcing term contains a Dirac sequence. Let (E-{S)f=
5(\,j).
Then J
The solution of
/(X)
11 _=/(o)^
a..
0co
x
1
+
(E - 0 ) / =
10
x
-
w
0co'
.
7
( x , i - f
g(\)'5(\J)
1).
X
/ ( X ) = /3 /(0) + ^ ( J > ( X , J
+
1).
11. The Scope of Generalized Sequences. Let / and g be sequences and define h by / = gh. If g(0) 9* 0, then h is a sequence whose members may be recursively calculated by 7=0
Let £ ^ 0. Then there is an element g(i) ^ 0. Therefore OO g =
OO
E 0 0 V i=%
= «E
g{jW~ l
j=i
Let e = the 2-transform (Kaplan [14, pp. 375-388], Lawden [1G])
/(*) = £'', y=O
and the generatrix transform (Riordan [20, Chapter 2]) fit)
=
E aij)t\ Y=O
It is clear that these transforms are closely related. The solution of linear difference equations with constant coefficients has been frequently performed by operational means. The reader is referred to Boole [3, Chapter XI], Jordan [13, Chapter XI], Milne-Thomson [18, Chapter XIII], Mikusinski [17, pp. 158-165] and Erdelyi [9, pp. 70-74]. The idea of multiplication defined by convolution is not new. The reader is referred to an important paper by Bell [1]. The key to Mikusinski's work is the introduction of an inverse to convolution. Moore [19] defines a convolution algebra in order to find formal power series solutions to differential equations. Feldman [10] and Elias [8] solve difference equations. Elias defines h = fg by
WIDE GAL
WIDE GAL P—65
f—65
Gal. 10 MT 3170 p. 44 Take l-14-10gx 18—1-5-65 *(X) = E / ( X -
- 1),
j)g(j
X > 0,
62 *(0) = 0.
Berg [2, p. 26] defines h = fg by
MX) = E / 0 ) ( x - j) - E / ( i ) i ? ( x - i - j). y*o
f f
y—o
In a paper published in 1950, Frankel [12] discusses a "calculus of figurate numbers." Frankel does not seem aware of the fact that his calculus is based on the sequences C(X, j) and (-l) C(j, X). Thus a number of authors have written on convolution algebras with applications to specific problems. The systematic application of such an algebra to a wide variety of problems in the discrete calculus is the contribution of this paper. We now investigate the relation between generalized sequences and generating functions. From (3.4), any sequence may be written as x
f = i f u w . y-o
(12.D
To the sequence with elements f(j) we may assign the generating function
= f:/oy.
/a)
(12.2)
y-o
Note that (12.1) is an identity which follows from the definition of the sequence co. It is simply a way of exhibiting the sequence in terms of its members and is always true. There can be no question of convergence. Equation (12.2) defines a mapping from a sequence to a function of a real variable. There is an obvious isomorphism between the two systems.
The conceptual difference between generating functions and generalized sequences may be illustrated by the following example. The generating function of the sequence 1, a, a , • • • is 1/(1 — at) because 2
I — at
y-o
On the other hand, 2
1 — aco
= 1, a, a ,
by the definition of fraction in our convolution algebra and by the definition of co. In the calculus of generalized sequences the formula for the coefficients of a product polynomial in terms of the coefficients of its factors is built into the system. In a sense, generalized sequences are to generating functions as the method of detached coefficients (synthetic division) is to polynomial division. On the other hand, the relations discovered during almost two hundred years of experience with generating functions may be used in the theory of generalized sequences because of the isomorphism. A p p e n d i x A . On Partial Fraction Expansions. The expansion of rational functions into partial fractions is too well known to require discussion here. The reader may refer to Kaplan [14, pp. 158-160]. The object of this appendix is to point out a simplification in the calculation of the coefficients. Let P(x)/Q(x) be a proper rational function, and let Q(x) have a zero a of degree p. Let
=
(A.1)
(*Q(x)
Then Q(x)
X -
a ^ (x -
y
a
+
^ (x -
a)" ^
K
h
where §
(A.2)
i = 0,l,
=
...,p-l,
PL,
and where \(x) is a partial fraction expansion determined by the other zeros of Q(x). From (A.l) and (A.2), it appears that the coefficients in the expansion of P(x)/Q(x) require the differentiation of a quotient which depends on P(x) and Q(x). We shall demonstrate that if the coefficients in the expansion of 1/Q(x) are Aj, then the Bj are linear combinations of the Aj with coefficients depending on P(x) only. The proof is simple. Let 1
=
Q(x)
Ai x -
A a
(x -
,
2
a)
2
, "'
A
^ (x -
p
a)*
'
T
where
Q(*) Since *>(x) = P(x)Kx),
fc-0 and 3
P^""*'^^
(A.3)
Equation (A.3) is the required relation. Partial fraction expansions occur in the solution of difference equations. In this case, Q(x) is determined only by the coefficients of the homogeneous equation, and P(x) is determined only by the initial conditions and the forcing term. Because of (A.3), the expansion due to the homogeneous equation may be determined and the initial conditions and the forcing term may then be varied without requiring the recalculation of the expansion coefficients from scratch. A p p e n d i x B. Automation of Operator Calculus. The solution of a problem with the aid of an operational calculus bears a resemblance to the process of solving a problem on a computer. First there is a translation phase. For a problem where an operational calculus is used this is typically a translation from an analytic to an algebraic problem. For a computer problem, this is the translation of, say, a mathematical formulation in the language of F o r t r a n into machine language. Then the problem is solved. Finally, an inverse translation is needed to make the results available. For the operational calculus, this involves the application of inverse transforms. For a computer problem this might mean binary-to-decimal conversion or conversion to c r t output. One advantage of an operational calculus lies in the fact that results may be obtained in a mechanical or "cookbook" fashion. This suggests that certain classes of problems could conveniently be solved on computers with an operational calculus providing a natural symbolism. Tables of transforms and inverse transforms would perform the functions of "syntax tables" in compilers. To change from, say, a
calculus for difference equations to one for differential equations, it is only necessary to change these tables.
The computer can also function as a repository of knowledge. For this purpose an operational calculus plays a natural role since the contents of certain parts of mathematics may be summarized in the algebraic structure of appropriate operators and in the transform tables. Some work on symbolic manipulation on digital computers has already been done. A noteworthy example is Brown's [4] a l p a k program which performs symbolic manipulation on rational functions of several variables. The symbolic solution of various types of linear systems seems a particularly suitable candidate for automation. Bell Telephone Laboratories, Incorporated Murray Hill, N e w Jersey
T
WIDE GAL P—72 GAL. LLX MT-APRIL 3170 P. 51, TAKE 1-14-10E, 319, 1-4-65.
1. E. T. BELL, "EULER ALGEBRA," Trans. Amer. Math. Soc, V. 25, 1923, PP. 13&-154. 2. L. BERG, EinfUhrung in die Operatorenrechnung,/FTIATHEMATIK FIIR NATURWISSENSCHAFT UND TCCHNIK, BD. 6, VEB DEUTSCHER VERLAG DER WISSENSCHAFTEN, BERLIN, 1962. MRFI5#5347.
3. G. BOOLE, Calculus of Finite Differences, 4TH ED., CHELSEA, NEW YORK, 1957. M1120 #1124. 4. W. S. BROWN, J. P. HYDE, & B . A. TAOUE, "THE ALPAK SYSTEM FOR NONNUMENCAL ALGEBRA ON A DIGITAL COMPUTER. II, RATIONAL FUNCTIONS ON SEVERAL VARIABLES," Bell System Tech. J., V. 43, 1964, PP. 785-304. _ 5. R. V. CHURCHILL, Operational Mathematics, MCGRAW-HILL, NEW YORK, 1958. MI$M#7411. 6. J. L. B . COOPER, "HEAVISIDE AND THE OPERATIONAL CALCULUS," Math. Gaz., V. 36,'1952, PP. 5-19. MIJ13, 612. 7. H. T. DAVIS, The Theory of Linear Operators, PRINCIPIA PRESS, BLOOMINGTON, INDIANA, 1936. 8. J. ELIXS, "ON AN OPERATOR METHOD FOR DIFFERENCE EQUATIONS," (CZECH), Mat.-Fys. Casopis
Sloven. Akad. Vied., V. 8, 1958, PP. 203-227. M^&L #4PD&
9. A. ERDELYI, Operational Calculus and Generalized Functions, HOLT, RINEHART AND WINSTON, NEW YORK, 1962. , 10. L. FELDMAN, "ON/FTINEAR DIFFERENCE EQUATIONS WITH CONSTANT COEFFICIENTS," Periodica
Polytech. Elec. Engrg., V. 3, 1959, PP. 247-257.
11. T. FORT, "LINEAR DIFFERENCE EQUATIONS AND THE DIRICHLET TRANSFORM," Amer. Math.
Monthly, V. 62, 1955, PP. 641-345. 12. E . T . FRANKEL, "A CALCULUS OFFIGURATENUMBERS ANDFINITEDIFFERENCES," Amer. Math. Monthly, V. 57, 1950, PP. 14-25. MRJLL, 412.
13. C. JORDAN, Calculus of Finite Differences, CHELSEA, NEW YORK, 1950. 14. W. KAPLAN, Operational Methods for Linear Systems, ADDISON-WESLEY SERIES IN MATHEMATICS, ADDISON WESLEY, READING, MASSACHUSETTS ^NXH'ALO ALTO, CALIFORNIA, 1962. MH25#3330. 15. K. S. KUNZ, Numerical Analysis, MCGRAW^IFLT,"NEW YORK, 1957. MR)19, 460. 16. D . F . LAWDEN, "ON THE SOLUTION OF LINEAR DIFFERENCE EQUATIONS," Math. Gaz.. V. 36, 1952, PP. 193-196. MTYL4, 285. 17. J. MIKUSINSKI, Operational Calculus, INTERNATIONAL SERIES OF MONOGRAPHS ON PURE AND APPLIED MATHEMATICS, VOL. 8, PERGAMON, NEW YORK, PANSFCF?OWE WYDAWIINICTIVO, NAUKOIVE, WARSAW,,^. MIJ21#4333. J
18. L. M. MILNE-THOMSON, The Calculus of Finite Differences, MACMILLAN, LONDON, 1933. 19. D . H. MOORE, "CONVOLUTION PRODUCTS AND QUOTIENTS AND ALGEBRAIC DERIVATIVES OF
SEQUENCES," Amer. Math. Monthly, V. 69. 1962, PP. 132-138. MI]P5# 402. 20. J. RIORDAN, An Introduction to Combinatorial Analysi^gWiley PUBLICATIONS IN MATHE-
MATICAL STATISTICS, WILEY, NEW YORK AND CHAPMAN