Introduction. The integral operators we shall be concerned with can be described with the help of the following:

Definition. A function F(u, v) will be called a primitive, if it is defined on the square 0 ≥ u, v ≥ 1, and satisfies the conditions (i) F(u, v) = 0 if u ≥ v; (ii) F(u, v) is a.c. (absolutely continuous) in each variable; (iii) there is a function p ∈ L(0, 1) such that |F1(u, v)| ≤ p(u), |F2(u, v)| ≤ p(v)a.e. (almost everywhere) on the square, where F1(u, v), F2(u, v) denote the partial derivatives of F(u, v) with respect to u, v respectively. It is not difficult to see that conditions (ii), (iii) imply that F(u, v) is a.c. in each variable uniformly with respect to the other variable or, as we say, is equi-a.c. in each variable. Thus, in the above definition, condition (iii) can be replaced by (iii), there is a p ∈ L(0, 1) such that for each v ∈ [0, 1], |F1(u, v)| ≤ p(u) a.e. in [0, 1] and for each v ∈ [0, 1], |F2(u, v)| ≤ p(u)a.e. in [0, 1].

The series is said to be summable (L) to sif

where sn = a0 + a1 + … + an, and it is said to be summable (l) to s if

It is shown that for k ≥ 5 in every k-chromatic graph there is a set of k distinct vertices V1, …, Vk with the property that for i, j = 1, …, k and i ≠ j the graph contains the union of a set of 4 paths connecting Vi and Vj no two of which have any edge or any vertex besides Vi and Vj in common and k − 1 paths connecting Vi and Vj no two of which have any edge in common.

In a recent note(1) I proved that if μ(n) denotes the usual Möbius function, N denotes a fixed positive integer and if

then

where T runs through all natural numbers ≤ x which are not divisible by an Nth power. In the present paper I shall establish some further relations of this character and, in particular, I shall prove that if

where

then

Thus, in some respects, L(x) appears more regular than M(x), the sum over L(x/T) being multiplicative, whereas M(x1/N) is not.

Let {Sn} be the sequence of partial sums of the infinite seriesΣαn. Let {pn} be a sequence of constants real or complex and let us set

The sequence {tn} of Nörlund means (5) or simply (N, pn) means of the sequence {Sn} generated by the sequence of coefficients {pn} is defined by the following sequence -to-sequence transformation

The series ∑αn or the sequence {Sn} is said to be summable (N, pn) to the sum S, if

and is said to be absolutely summable (N, pn) or summable |N, pn|, if the sequence {tn} is of bounded variation, that is, the series ∑|tn − tn−1| is convergent (2).

It is generally conjectured that if α1, α2 …, αk are algebraic numbers for which no equation of the form

is satisfied with rational ri not all zero, and if K > 1 + l/k, then there are only finitely many sets of integers p1, p2, …, pkq, q > 0, such that

This result would be best possible, for it is well known that (1) has infinitely many solutions when K = 1 + 1/k. † If α1, α2, …, αk are elements of an algebraic number field of degree k + 1 the result can be deduced easily (see Perron (11)). The famous theorem of Roth (13) asserts the truth of the conjecture in the case k = 1 and this implies that for any positive integer k, (1) certainly has only finitely many solutions if K > 2. Nothing further in this direction however has hitherto been proved.‡

On Uniform Interpolation Sets

By J. P. Earl

University of Kent

In the paper referred to in the title (these Proceedings 62 (1966), 721–742) the value of a constant used in the statement of several of the theorems is incorrect. Due to the erroneous appearance of a factor ¼ρ2 in the proof of Theorem 3 (penultimate equation of section 5), the value, or range of values, of τ should in fact be 4/ρ2 times the stated value.

There are also some misprints which escaped earlier detection. In the reference of Theorem A d = 0 should read d = D and in the second line of Proof of Theorem 3 Un should be defined by Un ≡ |z − zn| ≤ Δ |zn|1−½ρ. The right-hand side of equation (5.2) should be multiplied by σM, N(z).

In some problems where sine, cosine or Bessel functions are involved (e.g. (l), p. 35 et seq.), one has to satisfy certain initial conditions so that one obtains relations of the type

where t is a real parameter such that |t| < 1, and ±αn(n = 1, 2, …) are real simple zeros of an even or odd integral function f(z), being z = x + iy. An's are coefficients to be determined and, of course, ±ω is a real number that is not a zero of f(z).

Let λ > 0 and let

If , we say that the series is summable (P, λ) to s.

Methods are developed for the computation of the complex zeros of (½z)−νJν(z) when the index ν is an arbitrary complex number. These methods, which do not require an explicit knowledge of the Jv(z), are susceptible to rapid numerical evaluation on a computer. Beyond the interest in the zeros in their own right, these methods now make feasible the use of the infinite product representation of Jν(z) for the rapid computation of Bessel functions of complex order.

Let f be a 2π-periodic function of the class L(−π,π). Put

We call, with Žuk(6), the quantity L(p)(h, f) the L-modulus of smoothness of order p of the function f. Žuk has recently obtained, in (5) and (6), generalizations of a number of classical results on the absolute convergence of Fourier series, as also on the order of Fourier coefficients by employing the concept of the L-modulus of smoothness which is obviously a more general concept than that of the modulus of continuity. It is the purpose of this note to prove a theorem on the almost everywhere convergence of Fourier series of f involving the concept of L(p)(h, f).

By introducing complex variable techniques in the theory of shallow shells, the problem is reduced to the solution of one fourth-order differential equation, and the displacement components and stress resultants and couples are expressed in terms of four complex potentials. Two of these complex potentials are those of thin plate theory, thus admitting of deducing the solution of a shallow shell problem from the limiting case of the circular flat plate. These results are applied to a case of non-rotational symmetry, a special case of which has practical significance in the design of tanks and boilers.

1. Making use of the familiar abbreviation

let us adopt a contracted notation for the generalized hypergeometric function AFB[x] and write

where (a) denotes the sequence of parameters

1. Given an infinite series ∑an, we consider its Nörlund transformation

where

It is know that if

then the (N, p)-mean reduces to the familiar (C, α)-mean.

In this paper we give a new generating function for the Jacobi polynomials. The result obtained is of general character and includes as particular cases some of the results given earlier by Carlitz(2), Salam(1), Manocha and Sharma (3,4).

In this paper we investigate certain generalized forms of third and fourth order non-homogeneous differential equations. Ezeilo (l) derived interesting results for the problem

.

We show similar results for:

,

and

.

An integral transformation is denned over a finite interval of the time domain. When the Laplace transform exists, the finite transform yields identical results. However, the finite transform is found to be considerably more general than the Laplace transform. It permits consideration of functions which are not of exponential order, leads to a simple scheme to determine system response, and is applicable to boundary-value problems.

The main aim of the present paper is to give a new generating function for a product of n-Jacobi polynomials. The result established in this paper is the extension of my formula (4).

1. The consequences of assuming that an integral function or its derivatives take integer values for some or all integer values of the variable have been widely investigated. The best-known result of this kind was obtained by Pólya(9,10) and Hardy(6) in 1920 and states that among all transcendental integral functions which assume integer values for all non-negative integer values of the variable that of least increase is the function 2Z (cf. also (7)). Analogous results relating to integers in the Gaussian field were proved by Fukasawa(3)in 1926 and Gelfond(4)in 1929, and theorems relating to the function and its derivatives are included in the works of Gelfond (5), Selberg(14) and Straus (15). (For further results and references see (1,2, 11).) The earlier of these investigations may be regarded as the genesis of the celebrated Gelfond-Schneidersolution to the seventh problem of Hilbert and the much important work arising therefrom.

In this paper we obtain generating functions of Jacobi polynomials. The results obtained are of general character and include as particular cases of the results given earlier by Bailey ((1), p. 102, Ex. 19) and Brafman (2).