Summations Involving Binomial Coefficients

On the approximation of functions by tanh neural networks

“…This proves the claimed estimate (28) also for the case where p = 2n is even, and therefore concludes the proof of (28). Next, we optimize (28) by choosing the optimal value of α. Lemma A.2 proves that the optimal choice is α = 1/s.…”

“…We first note that choosing h as in Lemma 3.1 implies that (29) max p≤s, p odd E p ≤ , which proves the statement for p odd, since (1 + α)/α ≥ 1. We will now prove (28) for even p using induction. First note that…”

Close Encounters with the Stirling Numbers of the Second Kind

“…Here is a simple observation (made also in [12]) -expanding the binomial ( ) m xk y + in (1.2) and changing the order of summation, we find that (1.2) is based on the more simple identity (1.1), (which, by the way, is entry (1.13) in [7]). In his paper [8] Henry Gould provides a nice and thorough discussion of identity (1.1), which he calls Euler's formula, as it appears in the works of Euler on n-th differences of powers.…”

“…The starting point is one interesting identity. From time to time, in different books and articles we find the strange evaluation, For instance, the following theorem (extending (1.1) a little bit) appeared in [12].…”

On some combinatorial identities and harmonic sums

“…where a and b are real numbers with b = 0, and P is a polynomial of degree n with leading coefficient a 0 . In 2009 Katsuura [29] proved that…”