Some congruences involving the \(p\)-adic gamma function and some arithmetical consequences (Q2751730)
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scientific article; zbMATH DE number 1665069
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Some congruences involving the \(p\)-adic gamma function and some arithmetical consequences |
scientific article; zbMATH DE number 1665069 |
Statements
29 January 2002
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\(p\)-adic gamma function
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binomial coefficients
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congruences
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Some congruences involving the \(p\)-adic gamma function and some arithmetical consequences (English)
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Let \(\Gamma_p\) be the \(p\)-adic gamma function, \(p>5\). The author proves the congruence NEWLINE\[NEWLINE \Gamma_p(p^rx)\equiv \Gamma_p(p^r)^x\left[ 1+\frac{x(x^2- 1)}3p^r\sum\limits_{\underset {(k,p)=1}{k=1}}^{p^r}\frac{1}k\right]\pmod{p^{5r}\mathbb Z_p} NEWLINE\]NEWLINE where \(r\) is a positive integer, \(x\in \mathbb Z_p\). This implies several new congruences for binomial coefficients.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00058].
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