Intereting Posts

Solve a linear system with more variables than equations
Looking for confirmation on probability questions
How to use the method of “Hensel lifting” to solve $x^2 + x -1 \equiv 0\pmod {11^4}$?
Mean of fractional part of $\log n$
Equivalence of the definition of the Subbasis of a Topology
complete residue system modulo $p$
Show that $(f_n)$ is equicontinuous, given uniform convergence
Proving that an additive function $f$ is continuous if it is continuous at a single point
Least power. Squares again
Proving $\sum_{k=0}^{n}k{n\choose k}^2 = n{2n-1 \choose n-1} $
Proof of a theorem saying that we can make a measurable function continuous by altering it by a set of arbitarily small measure
beginner's question about Brownian motion
Is every subgroup of a normal subgroup normal?
Distribution of Max(X_i) | Min(X_i), X_i are iid uniform random variables
Generalizing the trick for integrating $\int_{-\infty}^\infty e^{-x^2}\mathrm dx$?

Define three generalized hypergeometric functions (which differs in the starting numerator in blue):

$$A_k =\,_kF_{k-1}\left(\left.\begin{array}{c} 1,\frac{\color{blue}{k-2}}{k-1},\frac{k-1}{k-1},\dots\\ \frac{k+1}{k}, \frac{k+2}{k},\dots \end{array}\right| \frac{(k-1)^{k-1}}{k^k}\right)$$

$$B_k =\,_kF_{k-1}\left(\left.\begin{array}{c} 1,\frac{\color{blue}{k-1}}{k-1},\frac{k}{k-1},\dots\\ \frac{k+1}{k}, \frac{k+2}{k},\dots \end{array}\right| \frac{(k-1)^{k-1}}{k^k}\right)$$

$$C_k =\,_kF_{k-1}\left(\left.\begin{array}{c} 1,\frac{\color{blue}{k}}{k-1},\frac{k+1}{k-1},\dots\\ \frac{k+1}{k}, \frac{k+2}{k},\dots \end{array}\right| \frac{(k-1)^{k-1}}{k^k}\right)$$

- Proof of binomial coefficient formula.
- A proof of the identity $ \sum_{k = 0}^{n} \frac{(-1)^{k} \binom{n}{k}}{x + k} = \frac{n!}{(x + 0) (x + 1) \cdots (x + n)} $.
- Binomial Coefficients Proof: $\sum_{k=0}^n {n \choose k} ^{2} = {2n \choose n}$.
- Asymptotics of sum of binomials
- sum of binomial coefficients involving $n,p,q,r$
- Proving a special case of the binomial theorem: $\sum^{k}_{m=0}\binom{k}{m} = 2^k$

The smallest common non-trivial case would then be $k=3$,

$$A_3={_3F_2}\left(1,\tfrac12,\tfrac22;\ \tfrac43,\tfrac53;\ \tfrac4{27}\right)$$

$$B_3={_3F_2}\left(1,\tfrac22,\tfrac32;\ \tfrac43,\tfrac53;\ \tfrac4{27}\right)$$

$$C_3={_3F_2}\left(1,\tfrac32,\tfrac42;\ \tfrac43,\tfrac53;\ \tfrac4{27}\right)$$

The first two were addressed in this and this posts by different OPs from different perspectives, but a common approach may be possible. First, let $x_n$ be the roots of $x^k-x+1=0$.

Question 1:

Is it true,

$$\frac{A_k}{(k-1)(k-2)} = \int_1^\infty \frac{-k+(k-1)x}{x^k-x+1}dx =-\sum_{n=1}^k x_n\ln(1-x_n)\tag1$$

**Note:** This implies a *simple* solution to the case $k=3$ discussed in by Reshetnikov in this post as,

$$\sum\limits_{n=0}^\infty\frac{n!\,(2n)!}{(3n+2)!}= -x_1\ln(1-x_1)-x_2\ln(1-x_2)-x_3\ln(1-x_3) = 0.5179778\dots$$

where the $x_n$ are the three roots of $x^3-x+1=0\;^\color{red}*$

and which is the minpoly of the negated *plastic constant*.

Question 2:

Is it true,

$$\frac{B_k}{k} = \int_1^\infty\frac1{x(x^k-x+1)}dx=-\sum_{n=1}^k \frac{\ln(1-x_n)}{-k+(k-1)x_n}\tag2$$

**Note:** This implies a similarly simple solution to the case $k=3$ in this post,

$$\sum_{n=1}^\infty\frac1{n\binom{3n}n} = \frac{\ln(1-x_1)}{3-2x_1}+ \frac{\ln(1-x_2)}{3-2x_2}+ \frac{\ln(1-x_3)}{3-2x_3}=0.371216\dots$$

and the $x_n$ again are the three roots of $^\color{red}*$.

Question 3:

See this post.

- Calculate $\int \limits {x^n \over 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+…+\frac{x^n}{n!}} dx$ where $n$ is a positive integer.
- The problem of instant velocity
- Verifing $\int_0^{\pi}x\ln(\sin x)\,dx=-\ln(2){\pi}^2/2$
- Product of Gamma functions II
- Usage of dx in Integrals
- Combinatorial interpretation of Binomial Inversion
- Preparing for Spivak
- Correct way to calculate numeric derivative in discrete time?
- Test for convergence the series $\sum_{n=1}^{\infty}\frac{1}{n^{(n+1)/n}}$
- How to evaluate $\int_0^1 \frac{\ln(x+1)}{x^2+1} dx$

- Do $\omega^\omega=2^{\aleph_0}=\aleph_1$?
- Dual space of the space of finite measures
- Prove (or disprove) the equivalence of AP, LUB, NIT and MCT
- Convergent sequence in Lp has a subsequence bounded by another Lp function
- Primary ideals in Noetherian rings
- Connectedness of points with both rational or irrational coordinates in the plane?
- Does taking the direct limit of chain complexes commute with taking homology?
- A question on morphisms of fields
- Prove sum of combinations
- Find X location using 3 known (X,Y) location using trilateration
- How can we conclude $n\xi_j^{n-1}=\prod_{i \ne j}(\xi_{i}-\xi_{j})$ from $x^n-1=\prod_{i}(x-\xi_{i})$
- Rank product of matrix compared to individual matrices.
- Equality of measures on a generated $\sigma$-algebra
- Looking for confirmation on probability questions
- If a, b ∈ Z are coprime show that 2a + 3b and 3a + 5b are coprime.