Finite and Infinite Geometric Series with Binomial Coefficients

Algebraic Polynomials with Random Coefficients with Binomial and Geometric Progressions

The expected number of real zeros of an algebraic polynomial ao a1x a2x · · · anx with random coefficient aj , j 0, 1, 2, . . . , n is known. The distribution of the coefficients is often assumed to be identical albeit allowed to have different classes of distributions. For the nonidentical case, there has been much interest where the variance of the jth coefficient is var aj ( n j ) . It is sh...

Binomial Coefficients Involving Infinite Powers of Primes

If p is a prime and n a positive integer, let νp(n) denote the exponent of p in n, and up(n) = n/p νp(n) the unit part of n. If α is a positive integer not divisible by p, we show that the p-adic limit of (−1) up((αp)!) as e → ∞ is a well-defined p-adic integer, which we call zα,p. In terms of these, we then give a formula for the p-adic limit of ( ap+c bpe+d ) as e → ∞, which we call ( ap∞+c b...

Series with Central Binomial Coefficients, Catalan Numbers, and Harmonic Numbers

We present several generating functions for sequences involving the central binomial coefficients and the harmonic numbers. In particular, we obtain the generating functions for the sequences ( 2n n ) Hn, ( 2n n ) 1 nHn, ( 2n n ) 1 n+1Hn , and ( 2n n ) n m. The technique is based on a special Euler-type series transformation formula.

Linear Differential Systems with Infinite Power Series Coefficients (Invited Talk)

On Certain Series Involving Reciprocals of Binomial Coefficients

We evaluate the following family of series involving reciprocals of binomial coefficients in terms of elementary functions for m = 3,4. ∞ ∑ k=0 xk (mk+1) (mk k ) .