Download Full text

A REMARK ABOUT THE BINOMIAL TRANSFORM
Massimo Galuzzi
Dipartimento di Matematica, Universita di Milano, Italy
(Submitted October 1996)
In [1, p. 137], Knuth introduced the idea of the binomial transform*
Given a sequence of numbers (a„), its binomial transform (a„) may be defined by the rule
*.=z.kh
kn-
(i)
Jc=0
Denoting the respective generating functions of (an) and {an) by A(x) and A(x), relation (1)
corresponds to
l-X
Vl-JC
(2)
In [2], Prodinger gives the following generalization. A sequence (an) may be transformed
into (an) by the rule
^mYkck-ak,
(3)
k=0
which corresponds to
1 .A[ cx
l-bx
[\-bx
A(X):
(4)
Now we may look at equation (4) as the action of a group structure over the set of functions.
Let C denote the field of complex numbers and C* denote the set of complex numbers different from 0. We define a group structure in C x C* by the law
(h, c) o (£', C') = (A' + bc\ cc').
(5)
Now
l-b'x
l__b
Cx
l-b!x
1
l-(b'+bc')x
\
cx
f
l-b x
\-b cxf
l-b xj
(6)
cc'x
{l-(h'+hc')x
Relation (6) shows that the action of the element (b, c) on A(x) followed by the action of the
element (b', c!) corresponds to the action of the product (bf + bc'y cc') over A(x).
It is easy to verify that the operation in (5) is associative. The unit element is given by (0,1),
and the inverse of the element (b,c) is given by (-h/c, lie). We immediately deduce that the
inversion formula for the binomial transform is
* A slight modification has been introduced in the original definition.
1998]
287
A REMARK ABOUT THE BINOMIAL TRANSFORM
It may be observed that the same group structure works equally well with the transformation
(ford fixed),
A(x) ->
v }
X
—r A f - ^ - 1 ,
(\-hx)d
\l-bxy
introduced by Prodinger in [2], corresponding to
REFERENCES
1. D. E. Knuth. The Art of Computer Programming 3. Reading, MA: Addison Wesley, 1973.
2. H. Prodinger. "Some Information about the Binomial Transform." The Fibonacci Quarterly
32.5(1994):412-15.
AMS Classification Number: 05A10
288