/pdf/an-approximation-theorem-for-the-poisson-binomial-2fgyl0mru1.pdf

An approximation theorem for the Poisson binomial distribution.

Substituting A=ydj(dy) for n, supposing the left member a polynomial in n, we construct the partial differential operator L=L(x, d\{dx), A) on functions of two independent variables. This operator is independent of n and is commutative with A. A solution of the simultaneous equation Lu=Q, Au=nu, where n is a constant, has the form u=vn(x)yn, where v=vn(x) is a solution of (1.1). Conversely, if v=vn(x) is a solution of (1.1), then ιt=vn(x)yn is a solution of the equations Lu = 0, Au = nu. Now suppose that, independently of the preceding considerations, we have obtained an explicit solution u=g(x, y) of Lu = 0, and that from the properties of this function we know that it has an expansion in powers of y of the form

https://projecteuclid.org/download/pdf_1/euclid.pjm/1172000968

Group-theoretic origin of certain generating functions

/pdf/functions-of-exponential-type-not-vanishing-in-a-half-plane-48xqchzxr6.pdf

Functions of exponential type not vanishing in a half-plane and related polynomials

/pdf/on-the-divisibility-of-the-class-number-of-quadratic-fields-1dfjyg7x3g.pdf

On the divisibility of the class number of quadratic fields

(1.5) If'(x)l _ MT12. In this paper we develop a unified method for arriving at these inequalities. In spite of being extremely simple, the method turns out to be very useful and effective. Not only does it give simpler proofs of the above results but yields interesting generalizations as well. We use the same idea to prove some other inequalities later in the paper. 2. The following result is well known (for references see [2, p. 82]). LEMMA 1. If f(z) is regular and of exponential type in the upper half plane, hf(7T/2)_c, and lf(x)_<M, -ooD<x<oo, then jf(x+iy) _Mecy, -soX<x0<oo, O_y<co. LEMMA 2. Let +(z) be an entire function of exponential type T which does not vanish in the upper half plane and h0,7Tr/2) = 0. If 0(z) = ei"zz(z), then for Im z < 0 (2.1) (Z)I -? I(Z))I andfor all real x (2.2) WI'(x) <WI'(x)l.

/pdf/functions-of-exponential-type-5cexpd935r.pdf

Functions of exponential type

/pdf/on-a-theorem-of-s-bernstein-22giyx926e.pdf

On a theorem of S. Bernstein

/pdf/criterion-for-rth-power-residuacity-2wtzp68uap.pdf

Criterion for rth power residuacity

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On a theorem of S. Bernstein

On the divisibility of the class number of quadratic fields

Criterion for rth power residuacity