L. Carlitz

Enumeration of pairs of sequences by rises, falls and levels

The paper is concerned with the enumeration of pairs of sequences with given specification according to rises, falls and levels. Thus there are nine possibilities RR, ..., LL. Generating functions in the general case are very complicated. However in a number of special cases simple explicit results are obtained.

Some polynomials related to theta functions

This paper continues the study of Hn(x)=Hn(x, q)=\(\Sigma _0^n \left[ {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right]x^r\) and related polynomials discussed bySzego [7]. In particular, it is shown that Hn(x) has various formal properties analogous to properties of theHermite polynomials.

Irreducibles and the composed product for polynomials over a finite field

Abstract Let GF(q) denote the finite field of q elements and let GF[q,x] denote the integral domain of polynomials in an indeterminate x over GF(q). Further, let Γ = Γ(q) denote the algebraic closure of GF(q) so that every polynomial in GF[q,x] on which there is defined a binary considers certain sets of monic polynomials from GF[q,x] on which there is defined a binary operation called the composed product. Here, if f and g are monics in GF[q,x] with deg f = m and deg g=n, then the composed product, denoted by f♦g and defined in terms of the roots of f and g, is also in GF[q,x] and has degree mn. In the present paper, the two most important composed products, denoted by the special symbols Ō and ∗, are those induced by the field multiplication and the field addition on Γ and defined by: f∘g = Π Π αβ (x − αβ), f∗g = Π Π αβ (x − (α+β)) , where the products indicated by П are the usual products in Γ[x] and are taken over all the roots α of f and β of g, (including multiplicities). These two composed products are called composed multiplication and composed addition, respectively. After introducing and developing some theory concerning a more general notion of composed product, this paper moves to the special composed products above and asks whether the irreducibles over GF(q) can be factored uniquely into indecomposables with respect to each of these products. Here, the term “irreducible” is used in the usual sense of the word while the term “indecomposable” is used in reference to composed products. This question is shown to have an affirmative answer in both situations, and thus yield unique factorization theorems (multiplicative and additive) for Γ. These theorems are then used to prove corresponding unique factorization theorems for all subfields of Γ. Next, it is shown that there are no irreducibles f in GF[q,x] which can be decomposed as f=f 1 O g 1 =f 2 ∗g 2 (except for trivial decompositions). A special inversion formula is then derived and using this inversion formula, the authors determine the numbers of irreducibles of degree n which are indecomposable with respect to (i) composed multiplication Ō, (ii) composed addition ∗, and (iii) both the composed products Ō and ∗ simultaneously. These numbers are given in terms of the well-known number of irreducibles of degree n over GF(q). A final section contains some discussion and several observations about the more general composed product.

Scalar polynomial functions on the n×n matrices over a finite field

Let F=GF(q) denote the finite field of order q, and let ƒ(x)ϵF[x]. Then f(x) defines, via substitution, a function from Fn×n, the n×n matrices over F, to itself. Any function ƒ:Fn×n → Fn×n which can be represented by a polynomialf(x)ϵF[x] is called a scalar polynomial function on Fn×n. After first determining the number of scalar polynomial functions on Fn×n, the authors find necessary and sufficient conditions on a polynomial ƒ(x) ϵ F[x] in order that it defines a permutation of (i) Dn, the diagonalizable matrices in Fn×n, (ii)Rn, the matrices in Fn×n all of whose roots are in F, and (iii) the matric ring Fn×n itself. The results for (i) and (ii) are valid for an arbitrary field F.

Note on irregular primes

where Bm denotes a Bernoulli number in the even-suffix notation. Jensen has proved that there exist infinitely many irregular primes of the form 4n+3 (for the proof see [3, p. 82]; see also [4]). In this note we give a simple proof of the weaker result that the number of irregular primes is infinite. We also prove a like result corresponding to the prime divisors of the Euler numbers. The letter p will always denote a prime >2.

A note on Euler numbers and polynomials

Let E m denote the Euler number in the even suffix notation so that (1.1) where, as usual, after expansion of the left member E r is replaced by E r . Nielsen [4, p. 273] has proved that (1.2)

Generating functions for certain types of permutations

Generating functions are obtained for certain types of permutations analogous to up-down and down-up permutations. In each case the generating function is a quotient of entire functions; the denominator in each case is φ02(x) − φ1(x)φ3(x), where φj(x)=∑n=o∞x4n+j(4n+j)!.

A note on the enumeration of line chromatic trees

Abstract Explicit formulas are obtained for the number of line chromatic planted trees with c line colors, n+1 lines and a given color on the stem, also for the number of rooted line chromatic trees with n lines at the root.

A NOTE ON THE MULTIPLICATION FORMULAS FOR THE BERNOULLI AND EULER POLYNOMIALS

where Bm(x), Em(x) denote the polynomials of Bernoulli and Euler in the usual notation. It is perhaps not so familiar that (1.1) and (1.2) characterize the polynomials. More precisely, as Nielsen has pointed out [3, p. 54], if a normalized polynomial satisfies (1.1) for a single value k> 1, then it is identical with Bm(x); similarly if a normalized polynomial satisfies (1.2) for a single odd k> 1, then it is identical with Em(x). For some generalizations see [1]. The situation for (1.3) is clearly different. For consider the equation

A test for additive decomposability of irreducibles over a finite field

Abstract A polynomial h over a field F is said to be additively decomposable over F if there exist polynomials f and g in F [ x ] each of degree >1 such that the roots of h are precisely all sums α + β of roots α of f and β of g . This paper derives a test for determining whether or not a given irreducible over a finite field is additively decomposable.