论文标题
一些新的系列以$ 1/π$的一致性动机
Some new series for $1/π$ motivated by congruences
论文作者
论文摘要
在本文中，我们以$ 1/π$的价格推论了一个六个新系列的家庭；例如，$$ \ sum_ {n = 0}^\ infty \ frac {41673840N+47777111} {5780^n} w_n \ left（\ frac {1444} = \ frac {147758475} {\ sqrt {95} \，π} $$其中$ w_n（x）= \ sum_ {k = 0}^n \ binom nk \ binom {n+k} k \ binom {2k} k \ binom {2（n-k）} {n-k} x^k $。为此，我们将我们的系列转变为一系列$$ \ sum_ {n = 0}^\ infty \ frac {an+b} {m^n} \ sum_ {k = 0}^n \ binom nk^4 $$ 2012年由库珀（Cooper）研究。例如，我们猜测$$ \ sum_ {k = 0}^\ infty \ frac {4290k+367} {3136^k} \ binom {2k} kt_k（14,1）t_k（17,16） $ x^k $在$（x^2+bx+c）^k $的扩展中。
In this paper, we deduce a family of six new series for $1/π$; for example, $$\sum_{n=0}^\infty\frac{41673840n+4777111}{5780^n}W_n\left(\frac{1444}{1445}\right) =\frac{147758475}{\sqrt{95}\,π}$$ where $W_n(x)=\sum_{k=0}^n\binom nk\binom{n+k}k\binom{2k}k\binom{2(n-k)}{n-k}x^k$. To do so, we transform our series to series of the type $$\sum_{n=0}^\infty\frac{an+b}{m^n}\sum_{k=0}^n\binom nk^4$$ studied by Cooper in 2012. In addition, we pose $17$ new series for $1/π$ motivated by congruences; for example, we conjecture that $$\sum_{k=0}^\infty\frac{4290k+367}{3136^k}\binom{2k}kT_k(14,1)T_k(17,16)=\frac{5390}π,$$ where $T_k(b,c)$ is the coefficient of $x^k$ in the expansion of $(x^2+bx+c)^k$.
