1. Guinand (2) has obtained finite identities of the type

where m, n, N are positive integers and either

or

where γ is Euler's constant and the notation ∑′ indicates that when x is integral the term r = x is multiplied by ½. Clearly there is no loss of generality in taking N = 1 in (1.1).

Leeming [4] has defined a sequence of polynomials {Q 4n (x)} and a sequence of integers {Q 4n } by means of

1

and

2

Thus

3

Leeming showed that the Q 4n are all odd and that

4

It is proved in [3] that

5

Leeming [3] has defined a sequence of polynomials {Q 4n (x)} and a sequence of integers {Q4n } by means of

1.1

and

1.2

Thus

1.3

and equivalently

1.4

Let A: a1 < a2 < … < ak be a set of non-negative integers We call the corresponding polynomial the characteristic polynomial, or briefly, the c-polynomial of A. Any polynomial of such a form we call a c-polynomial and any factorization of a c-polynomial into others of the kind we call a c-factorization. If a c-polynomial cannot be factored in this way we call it c-irreducible. In this note we will determine all c-factorizations of the polynomial 1 + x + x2 + … + xn-1, and will find under what circumstances the c-irreducible factors are also irreducible in the usual sense, i.e., irreducible over the field of rationals.

It is well known that the number of normalized irreducible polynomials of degree m in a single indeterminate, with coefficients in GF(q), is given by

(1.1)

where μ(r) is the Möbius function. It follows from (1) that

(12)

Let and Hn(x) be the nth Laguerre and Hermite polynomials, respectively. Two well-known bilinear generating formulas are the Hardy-Hille formula [1, p. 101]

and the mehler formula [1, p. 377]

This suggests the following problem. Consider the equation

where fa(x) is a polynomial in x of degree n with highest coefficient equal to 1,

A0 = B0 = 1. We shall also assume that ak = 1 and y0y1y2 … yk–1 ≠ 0. We seek all sets of polynomials {fn(x)} which satisfy (1.3), (1.4) and (1.5).

Let Fn (k) denote the total number of k-coloured graphs on n labelled nodes and let Mn (k) denote the number of graphs on n nodes that are coloured in at most kcolours ; also let fn (k) denote the number of connected k coloured graphs on n nodes. Read (3) has proved the following formulas:

Put

In a recent paper (4), Lohne showed that

Let p be a prime and let F be a polynomial in one variable with coefficients in GF(p), the field of p elements. Let d be the degree of F, and let r+1 denote the number of distinct values F(µ) as µ. ranges over GF(p). A generalization of the Waring problem modulo p leads to the question the determination of a lower bound for r.

Let

It is familiar that

(1) In the addition theorem [3, p. 363]

where

take θ = π and replace v by v−½. Since

we get

The formula

was stated by Bateman ([2], p. 457); a proof is sketched in [3], p. 144. Here

the Laguerre polynomial of degree

1. Good (4) has proved the formula

where Pn(x) is the Legendre polynomial of degree n and t is any integer greater than n. The form of the right-hand side suggests that (1) may be of use in deriving arithmetic properties of Pn(x).

Elsewhere (1) the writer indicated a connexion between divisibility properties of Pm(a) for special values of a and the complex multiplication of elliptic functions. If p = 2m + 1 is an odd prime, put

Let p be a prime > 2 and m an arbitrary positive integer; define

where (r/p) is the Legendre symbol. We consider the problem of finding the highest power of p dividing Sm. A little more generally, if we put

where a is an arbitrary integer, we seek the highest power of p dividing Sm(a). Clearly Sm = Sm(0), and Sm(a) = Sm(b) when a ≡ b (mod p).

Let F q = GF(q) denote the finite field of orderq = pn , where p is a prime. Consider the group Γ of linear transformations

(1.1) x' = (ax + b)//(cx + d)

with coefficients a, b, c, d ∊ Fq and of determinant 1. The order of Γ is½q(q 2 − 1) or q(q2 − 1) according as q is odd or even, i.e., according as p > 2 or p = 2.

Let Em denote the Euler number in the even suffix notation so that

(1.1)

where, as usual, after expansion of the left member Er is replaced by Er. Nielsen [4, p. 273] has proved that

(1.2)

For a given sequence {am} and p≠0, Schur (2) defined

In particular if p is a prime, a an integer and , then by Fermat's theorem

is integral. Schur proved that if p † a, then all the derivatives

are integral. Zorn (3) using p-adic methods proved Schur's results and also found the residue of Xm (mod pm), where and x = 1 (mod p). The writer (1) proved Zorn's congruences by elementary methods as well as certain additional results of a similar sort.