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Theorem 1.1 does not refer to a probabilistic limit, it holds elementwise for any given series of matrices that satisfies the conditions stated in the theorem.

$$I_{k,m}(n)=\int_{0}^{1}u^k\cot\left({\frac{\pi(1-u)}{m}}\right)\sin\left({\frac{2\pi n(1-u)}{m}}\right)\,du$$ $$\lim_{n\rightarrow\infty} I_{k,m}(n)=\frac{1}{2}m, \;\;\text{for}\;\; m\geq 1\;\; \te …

UPDATE I tried to evaluate the sum numerically for large $n$ and what I find does not support the conclusion I give below, that the large-$n$ limit equals 1 independent of $c$. Here is a plot for $c= …

The large-$x$ limit is only zero if $a+b>3/2$: $$_1{F}_2({1}; {a, b}; -x^2/4)=$$ $$=\sqrt{\tfrac{1}{\pi}}\Gamma (a) \Gamma (b) \sin \left(\tfrac{\pi}{2} (a+b-\tfrac{1}{2})-x\right)(2/x)^{a+b-3/2}+{\ …

Use the expansion $$\zeta(s)=\frac{1}{s-1}+\gamma+{\cal O}(s-1),$$ hence, since $\zeta(x)=1+2^{-x}+3^{-x}+4^{-x}+5^{-x}+\cdots$, you have $$\zeta(\zeta(x))=\frac{2^x}{1+(2/3)^x+(2/4)^x+(2/5)^x+\cdots} …

A series with precise error estimates is derived in Error bounds for a uniform asymptotic expansion of the Legendre function: $$P_n(\cosh x)=\left(\frac{x}{\sinh x}\right)^{1/2}\sum_{\nu=0}^{\infty}c …

Mathematica can actually solve the recursion relation in closed form, $$F_n(n)=-\tfrac{1}{2}(n^2-1)^{-1}\left(\frac{n-1}{n}\right)^n\left[n \left(\frac{n}{n-1}\right)^n \Phi \left(\frac{n}{n-1},1,n+1\ …

For each $k>0$, $c\in\mathbb{C}$ it holds that $$\lim_{\alpha\rightarrow\infty}\frac{\Gamma(\alpha+c)}{\Gamma(\alpha)\alpha^c}=1\Rightarrow\lim_{\alpha\rightarrow\infty} \left(\log\frac{\Gamma(k\alpha …

The sum $\sum _{n=0}^{\infty}t^{n^2}$ evaluates for $t<1$ to an elliptic theta function, and then taking the limit $t\rightarrow 1$ from below gives $$\lim_{t\nearrow 1}\sqrt{1-t}\sum _{n=0}^{\infty}t …

For any $\delta,x$ such that $0<\delta\leq |x|<\pi$ one has $|F_N(x)|\leq[2\pi(N+1)\sin^2(\delta/2)]^{-1}$, so the error in the delta-function approximation is of order $1/N$. Two explicit examples, i …

You can reduce the evaluation of $S_n$ to a quadrature by means of the Abel-Plana formula, $$S_n=\sum _{s=1}^{n-1} g(s)=\int_1^{n-1}g(s)\,ds+\tfrac{1}{2}g(1)+\tfrac{1}{2}g(n-1)$$ $$\qquad\qquad -\,2\o …

The OP asks where this recursion relation might appear in a research context. It appears as a discretization of the Emden–Fowler nonlinear differential equation, $$f''(t)=t^{p}[f(t)]^q,$$ for $p=1$, $ …

You might try to regularize the sum, $$S(\alpha)=\sum_{n=1}^\infty \frac{ \cos(\alpha n) \sin(n+1) }{n}= -\tfrac{1}{4} i \left[e^{-i} \ln \left(1-e^{i (\alpha-1)}\right)+e^{-i} \ln \left(1-e^{-i (\alp …

Since $K_0(a\sqrt{x})\rightarrow-\frac{1}{2}\ln x-\gamma_{\rm Euler}+\ln(2/a)$ for small $x$, one has $$\lim_{x\rightarrow 0}\; (x\ln^2 x)\frac{\partial }{{\partial x}}\frac{{{K_0}(a\sqrt x )}}{{{K_0} …

A general approach to obtain upper and lower bounds on $P(|X_1|\leq M_1, |X_2|\leq M_2,\dots, |X_d|\leq M_d)$ for a singular multivariate Gaussian, with a noninvertible covariance matrix, is developed …