Results in Mathematics ( IF 2.2 ) Pub Date : 2020-04-24 , DOI: 10.1007/s00025-020-01195-3 Xiaoxia Wang , Mingbing Yue
Recently, Jana and Kalita proved the following supercongruences involving rising factorials \((\frac{1}{d})_k^3\):
$$\begin{aligned}&\sum _{k=0}^{N} (-1)^k (2dk+1)\frac{(\frac{1}{d})_k^3}{k!^3}\\&\quad \equiv {\left\{ \begin{array}{ll} p^r&{}\pmod {p^{r+2}},\quad \text {if } r \text { is even};\\ (-1)^{\frac{(p-d+1)r}{d}}(d-1)p^r&{}\pmod {p^{r+2}},\quad \text {if } r \text { is odd}; \end{array}\right. }\\&\sum _{k=0}^{N} (-1)^k (2dk+1)^3\frac{(\frac{1}{d})_k^3}{k!^3}\\&\quad \equiv {\left\{ \begin{array}{ll} -3p^r&{}\pmod {p^{r+2}},\quad \text {if } r \text { is even};\\ (-1)^{\frac{(p+1)r}{d}}3(d-1)p^r&{}\pmod {p^{r+2}},\quad \text {if } r \text { is odd}, \end{array}\right. } \end{aligned}$$where \(N={\left\{ \begin{array}{ll} \frac{p^r-1}{d}, \quad &{}\text {if } r \text { is even};\\ \frac{(d-1)p^r-1}{d},\quad &{}\text {if } r \text { is odd}.\end{array}\right. }\) From Watson’s \(_8\phi _7\) transformation formula, we give q-analogues of the above supercongruences, generalizing some previous conjectural results of Van Hamme. Our proof uses the ‘creative microscoping’ method which was introduced by Guo and Zudilin.
中文翻译：
沃森的$$ _ 8 \ phi _7 $$ 8ϕ7转换公式的q-超同余
最近，Jana和Kalita证明了以下涉及乘数阶乘\（（\ frac {1} {d}）_ k ^ 3 \）的超同余：
$$ \ begin {aligned}＆\ sum _ {k = 0} ^ {N}（-1）^ k（2dk + 1）\ frac {（\ frac {1} {d}）_ k ^ 3} {k ！^ 3} \\＆\ quad \ equiv {\ left \ {\ begin {array} {ll} p ^ r＆{} \ pmod {p ^ {r + 2}}，\ quad \ text {if} r \文字{is even}; \\（-1）^ {\ frac {（p-d + 1）r} {d}}（d-1）p ^ r＆{} \ pmod {p ^ {r + 2} }，\ quad \ text {if} r \ text {是奇数}; \ end {array} \ right。} \\＆\ sum _ {k = 0} ^ {N}（-1）^ k（2dk + 1）^ 3 \ frac {（\ frac {1} {d}）_ k ^ 3} {k！^ 3} \\＆\ quad \ equiv {\ left \ {\ begin {array} {ll} -3p ^ r＆{} \ pmod {p ^ {r + 2}}，\ quad \ text {if} r \ text {is even}; \\（-1）^ {\ frac {（p + 1）r} {d}} 3（d-1）p ^ r＆{} \ pmod {p ^ {r + 2}}， \ quad \ text {if} r \ text {是奇数}，\ end {array} \ right。} \ end {aligned} $$其中\（N = {\ left \ {\ begin {array} {ll} \ frac {p ^ r-1} {d}，\ quad＆{} \ text {if} r \ text {is even}; \ \ \ frac {（d-1）p ^ r-1} {d}，\ quad＆{} \ text {if} r \ text {isodd}。\ end {array} \ right。} \）来自Watson's \（_ 8 \ phi _7 \）转换公式，我们给出上述超同余的q-模拟，归纳了Van Hamme先前的一些猜想结果。我们的证明使用了郭和祖迪林引入的“创意微观范围”方法。
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