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An extension of Lucas’ theorem
 Proc. Amer. Math. Soc
"... Abstract. Let p be a prime. A famous theorem of Lucas states that ( mp+s) ≡ ( np+t m) ( s) (mod p) ifm, n, s, t are nonnegative integers with s, t < p. Inthispaper n t we aim to prove a similar result for generalized binomial coefficients defined in terms of second order recurrent sequences with ..."
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Cited by 26 (16 self)
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Abstract. Let p be a prime. A famous theorem of Lucas states that ( mp+s) ≡ ( np+t m) ( s) (mod p) ifm, n, s, t are nonnegative integers with s, t < p. Inthispaper n t we aim to prove a similar result for generalized binomial coefficients defined in terms of second order recurrent sequences
A Refinement Of The GaussLucas Theorem
"... The classical Gauss  Lucas Theorem states that all the critical points (zeros of the derivative) of a nonconstant polynomial p lie in the convex hull \Xi of the zeros of p. It is proved that, actually, a subdomain of \Xi contains the critical points of p. 1. ..."
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Cited by 8 (0 self)
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The classical Gauss  Lucas Theorem states that all the critical points (zeros of the derivative) of a nonconstant polynomial p lie in the convex hull \Xi of the zeros of p. It is proved that, actually, a subdomain of \Xi contains the critical points of p. 1.
INVERSE SPECTRAL PROBLEM FOR NORMAL MATRICES AND THE GAUSSLUCAS THEOREM
"... Abstract. We establish an analog of the CauchyPoincare interlacing theorem for normal matrices in terms of majorization, and we provide a solution to the corresponding inverse spectral problem. Using this solution we generalize and extend the Gauss–Lucas theorem and prove the old conjecture of de B ..."
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Cited by 28 (0 self)
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Abstract. We establish an analog of the CauchyPoincare interlacing theorem for normal matrices in terms of majorization, and we provide a solution to the corresponding inverse spectral problem. Using this solution we generalize and extend the Gauss–Lucas theorem and prove the old conjecture of de
society THE GAUSSLUCAS THEOREM AND JENSEN POLYNOMIALS BY
"... Abstract. A characterization is given of the sequences {"fyj^o vvith the property that, for any complex polynomial/(z) = 1akzk and convex region Kcontaining the origin and the zeros of/, the zeros of 2 y¡<akzk again lie in K. Many applications and related results are also given. This work l ..."
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Abstract. A characterization is given of the sequences {"fyj^o vvith the property that, for any complex polynomial/(z) = 1akzk and convex region Kcontaining the origin and the zeros of/, the zeros of 2 y¡<akzk again lie in K. Many applications and related results are also given. This work leads to a study of the Taylor coefficients of entire functions of type I in the LaguerrePólya class. If the power series of such a function is given by 1 ykzk/k \ and the sequence {yk} is positive and increasing, then the sequence satisfies an infinite collection of strong conditions on the differences, namely A"yA> 0 for all n, k. 1. Introduction. This paper is concerned with functions of type I in the LaguerrePólya class; i.e. real entire functions which are the uniform limits, on compact subsets of the plane, of polynomials with only real zeros, all of which have the same sign. Let us represent such a function as a series <&(z) = "2k=0ykzk/k\. Pólya and Schur [PS] gave two alternate characterizations of this class of entire functions.
Bernoulli numbers, Wolstenholme’s theorem, and p 5 variations of Lucas’ theorem
 J. Number Theory
, 2007
"... Abstract. In this note we shall improve some congruences of G.S. Kazandzidis and D.F. Bailey to higher prime power moduli, by studying the relation between irregular pairs of the form (p, p − 3) and refined version of Wolstenholme’s theorem. 1 ..."
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Cited by 18 (4 self)
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Abstract. In this note we shall improve some congruences of G.S. Kazandzidis and D.F. Bailey to higher prime power moduli, by studying the relation between irregular pairs of the form (p, p − 3) and refined version of Wolstenholme’s theorem. 1
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