-
![]()
b. [1+] Let t(n) be the number of total partitions of n, as defined in Exam-
ple 5.2.5. Let g(n) have the same meaning as in Exercise 5.26. Deduce from
(a) that g(n) = 2"t(n) for n >1.
c. [2+] Give a simple combinatorial proof of (b).
5,37. a. [2+] Let 1=D po(x), pi(x), be a sequence of polynomials (with coeffi-
cients in some field K of characteristic O0), with deg pn=n for all nE N.
Show that the following four conditions are equivalent:
) Pn(x + y) =DE>o (") Pe(x)pnーk(y), for all n eN.
(i) There exists a power series f(u)=aju+azu'+ E K [[u]] such that
と P(x)-
un
expxf(u).
(5.110)
n!
n>0
仮定
NOTE: The hypothesis that deg pPn=nimplies that aj ¥ 0.
() E20 Pa(x) = (E>0 Pn(1)).
(iv) There exists a linear operator Q on the vector space K[x] of all poly-
nomials in x, with the following properties:
●Ox is a nonzero constant
●Qis a shift-invariant operator, i.e., for all aeK,Qcommutes with
the shift operator E4 defined by E® p(x)=D p(x +a).
● We have
Qpn(x) =D npn-1(x)
for all n e P.
NOTE: A sequence po, Pi, .. . of polynomials satisfying the above con-
ditions is said to be of binomial tvpe. The operator Q is called a delta
