22,300 research outputs found
On a conjecture of Wilf
Let n and k be natural numbers and let S(n,k) denote the Stirling numbers of
the second kind. It is a conjecture of Wilf that the alternating sum
\sum_{j=0}^{n} (-1)^{j} S(n,j) is nonzero for all n>2. We prove this conjecture
for all n not congruent to 2 and not congruent to 2944838 modulo 3145728 and
discuss applications of this result to graph theory, multiplicative partition
functions, and the irrationality of p-adic series.Comment: 18 pages, final version, accepted for publication in the Journal of
Combinatorial Theory, Series
Chebyshev Series Expansion of Inverse Polynomials
An inverse polynomial has a Chebyshev series expansion
1/\sum(j=0..k)b_j*T_j(x)=\sum'(n=0..oo) a_n*T_n(x) if the polynomial has no
roots in [-1,1]. If the inverse polynomial is decomposed into partial
fractions, the a_n are linear combinations of simple functions of the
polynomial roots. If the first k of the coefficients a_n are known, the others
become linear combinations of these with expansion coefficients derived
recursively from the b_j's. On a closely related theme, finding a polynomial
with minimum relative error towards a given f(x) is approximately equivalent to
finding the b_j in f(x)/sum_(j=0..k)b_j*T_j(x)=1+sum_(n=k+1..oo) a_n*T_n(x),
and may be handled with a Newton method providing the Chebyshev expansion of
f(x) is known.Comment: LaTeX2e, 24 pages, 1 PostScript figure. More references. Corrected
typos in (1.1), (3.4), (4.2), (A.5), (E.8) and (E.11
On the non-quadraticity of values of the q-exponential function and related q-series
We investigate arithmetic properties of values of the entire function that includes as
special cases the Tschakaloff function () and the -exponential
function (). In particular, we prove the non-quadraticity of the
numbers for integral , rational and
, .Comment: 27 page
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