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

Generalize $\frac1{48^{1/4}\,K(k_3)}\,\int_0^1 \frac{dx}{\sqrt{1-x}\,\sqrt[3]{x^2+27x^3}}=\,_2F_1\big(\tfrac13,\tfrac13;\tfrac56;-27\big)=\frac47\,$?

These involve separate cases $a=\tfrac13,\,a=\tfrac14,\,a=\tfrac16$ of, $${_2F_1\left(a ,a ;a +\tfrac12;-u\right)}=2^{a}\frac{\Gamma\big(a+\tfrac12\big)}{\sqrt\pi\,\Gamma(a)}\int_0^\infty\frac{dx}{(1+2u+\cosh x)^a}\tag1$$ The function $\eta(\tau)$ below is the Dedekind eta function. I. Case $a=\tfrac13$ Conjecture: There are an infinitely many algebraic numbers $\alpha, \beta$ such that $$H_1(\tau) =\,_2F_1\big(\tfrac13,\tfrac13;\tfrac56;-\alpha \big)=\beta$$ given by, $$\alpha = \frac1{4\sqrt{27}}\big(\lambda^3-\sqrt{27}\,\lambda^{-3}\big)^2$$ where $\lambda=\large{\frac{\eta(\frac{\tau+1}3)}{\eta(\tau)}}$ and $\tau=\frac{1+N\sqrt{-3}}2$ for any integer $N>1$. Examples: $$H_1\big(\tfrac{1+5\sqrt{-3}}2)=\,_2F_1\big(\tfrac13,\tfrac13;\tfrac56;-4 \big)=\tfrac3{5^{5/6}}$$ $$H_1\big(\tfrac{1+7\sqrt{-3}}2)=\,_2F_1\big(\tfrac13,\tfrac13;\tfrac56;-27 … Read more

Why are there elementary equations that are not solvable in closed form?

Elementary equations and closed-form solutions can be important in mathematics and in the natural sciences, e.g. for discovering and describing certain relationships. log:x↦log(x); x≠0; −π<ℑ(log(x))≤π for all x L denotes the Liouvillian numbers (= Elementary numbers). L is the smallest field that contains Q and is closed under algebraic operations, exp and log. The Elementary … Read more

Expressing the inverse Dixon function in terms of more familiar functions

If x3+y3−3αxy=1, is there an expression for the integral ∫z0dxy2−αx in terms of more familiar functions? A.C. Dixon introduced the elliptic functions sm(u,α) and cm(u,α) now named after him in this article. It can be shown (see e.g. my writeup here) that these functions can be expressed in terms of the more conventional Weierstrass elliptic … Read more

Two questions on asymptotic expansion of confluent hypergeometric functions for real variable x,|x|→∞x, |x| \to \infty

I’m looking into the asymptotic expansion for confluent hypergeometric function 1F1(a;b;z)≡M(a;b;z) and I’ve two quick questions regarding its asymptotic behavior for real values x, i.e. I’m interested in the asymptotic behavior of 1F1(a;b;x)≡M(a;b;x),x∈R,x→∞. A comment on notation: Below M(a;b;z),M(a;b;z) are indeed different and they’re related by M(a;b;z)=Γ(b)M(a;b;z),Γ(.) denoting the Gamma function, link below. (1) From … Read more

Is there a special function for ∑∞m=0xm/(m!)s\sum_{m=0}^{\infty} x^m/(m!)^s?

Is there a special function for the following series? ∞∑m=0xm(m!)s Here, s is a positive real number. When, s is an integer, s=n∈Z, this series can be written in terms of the generalized hypergeometric function: ∞∑m=0xm(m!)n=0Fn−1(1,⋯,1|x) . What is the analogue of the above for more general values of s, e.g. s=1/2? Answer AttributionSource : Link … Read more

Trigonometry and plane geometry

This will be a variation on the theme of this question, or maybe a rephrasing of it with a somewhat readjusted emphasis. In this posting I introduced the function \begin{align} & f_3(\theta_1,\theta_2,\theta_3,\ldots) \\[8pt] = {} & \sum_{\text{odd } n\,\ge\,3} (-1)^{(n-3)/2} (n-1) \sum_{|A|\,=\,n} \prod_{i\,\in\,A} \sin\theta_i\sum_{i\,\notin\,A} \cos\theta_i \tag{$f_3$} \end{align} and the conditional trigonometric identity $$ \text{If } … Read more