## Asymptotic expansion of Mellin transform of products of modified Bessel function K

Let n≥1 be an integer, let F(x,y)=∫∞0un(x+y)(Kx−y(u))ndu for x,y≥0. When n=1, this is just Mellin transform of the Bessel K function. When n=2, F(x,y) has an explicit form in product of Gamma functions, given by the Parseval formula for Mellin transform. For general n, I expect some Stirling formula type estimation for F(x,y). I tried … Read more

## numerical stability of root identification via Newton-Raphson iteration of Stieltjes residue sums

I have asked several questions on math.SE in order to compute numerically the poles of high-degree Padé approximations for e−x, because a computation directly from the polynomial coefficients has poor numerical stability. I finally blundered upon an approach alluded to a paper by Campos and Calderón and a related item mentioned in a paper by … Read more

## Functional equations about Conway’s box function

Conway’s box function is the inverse of Minkowski’s question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence). The question is are there functional equations known about the function, which would allow recursive computation? Answer AttributionSource : Link , Question Author : Dimiter P … Read more

## Christoffel-Darboux type identity

The classical Christoffel-Darboux identity for Hermite polynomials reads n∑k=0Hk(x)Hk(y)2kk!=12n+1n!Hn+1(x)Hn(y)−Hn(x)Hn+1(y)x−y. I am interested in a similar identity, where one of the indices is shifted by one, explicitly Fn(x,y):=n∑k=0Hk+1(x)Hk(y)2k+1(k+1)!=xn∑k=01k+1Hk(x)Hk(y)2kk!−n∑k=0kk+1Hk−1(x)Hk(y)2kk!, where the recursion relation Hk+1(x)=2xHk(x)−2kHk−1(x) was used. In the first term Christoffel-Darboux cannot be applied due to the 1/(k+1) prefactor and the second term is not quite Fn−1(y,x) … Read more