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p-adic gamma function

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Mathematics" title="Mathematics">mathematics</a>, the p-adic gamma function Γp is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> of a <a href="P-adic" class="mw-redirect" title="P-adic">p-adic</a> variable analogous to the <a href="Gamma_function" title="Gamma function">gamma function</a>. It was first explicitly defined by <a href="#CITEREFMorita1975">Morita (1975)</a>, though <a href="#CITEREFBoyarsky1980">Boyarsky (1980)</a> pointed out that <a href="#CITEREFDwork1964">Dwork (1964)</a> implicitly used the same function. <a href="#CITEREFDiamond1977">Diamond (1977)</a> defined a p-adic analog Gp of log Γ. <a href="#CITEREFOverholtzer1952">Overholtzer (1952)</a> had previously given a definition of a different p-adic analogue of the gamma function, but his function does not have satisfactory properties and is not used much.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

The p-adic gamma function is the unique continuous function of a <a href="P-adic_integer" class="mw-redirect" title="P-adic integer">p-adic integer</a> x (with values in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation>

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</math><img src="./dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}" loading="lazy">) such that

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{p}(x)=(-1)^{x}\prod _{0<i<x,\ p\,\nmid \,i}i}</annotation>

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</math><img src="./e0332c4816df80a00b523337da59660bd9b8a6b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:24.829ex; height:6.009ex;" alt="{\displaystyle \Gamma _{p}(x)=(-1)^{x}\prod _{0<i<x,\ p\,\nmid \,i}i}" loading="lazy"></dd></dl>

for <a href="Positive_integer" class="mw-redirect" title="Positive integer">positive integers</a> x, where the product is restricted to integers i not <a href="Divisibility" class="mw-redirect" title="Divisibility">divisible</a> by p. As the positive integers are dense with respect to the p-adic topology in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation>

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</math><img src="./dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{p}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{p}(x)}</annotation>

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</math><img src="./a2b486a1b0c62c859a949b1b19d1e9feef074a25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.651ex; height:3.009ex;" alt="{\displaystyle \Gamma _{p}(x)}" loading="lazy"> can be extended uniquely to the whole of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation>

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</math><img src="./dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}" loading="lazy">. Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation>

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</math><img src="./dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}" loading="lazy"> is the ring of <a href="P-adic_integer" class="mw-redirect" title="P-adic integer">p-adic integers</a>. It follows from the definition that the values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{p}(\mathbb {Z} )}">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{p}(\mathbb {Z} )}</annotation>

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</math><img src="./db0e150073c245e38e395457ba5a438cba22208c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.872ex; height:3.009ex;" alt="{\displaystyle \Gamma _{p}(\mathbb {Z} )}" loading="lazy"> are invertible in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation>

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</math><img src="./dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}" loading="lazy">; this is because these values are products of integers not divisible by p, and this property holds after the continuous extension to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation>

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</math><img src="./dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}" loading="lazy">. Thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{p}:\mathbb {Z} _{p}\to \mathbb {Z} _{p}^{\times }}">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{p}:\mathbb {Z} _{p}\to \mathbb {Z} _{p}^{\times }}</annotation>

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</math><img src="./637908adde176a3a9364d6aab7b778ea5f4ba56e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.734ex; height:3.176ex;" alt="{\displaystyle \Gamma _{p}:\mathbb {Z} _{p}\to \mathbb {Z} _{p}^{\times }}" loading="lazy">. Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}^{\times }}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}^{\times }}</annotation>

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</math><img src="./cad3eed5c24322475c3d9a215ef65badd48087f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.061ex; height:3.176ex;" alt="{\displaystyle \mathbb {Z} _{p}^{\times }}" loading="lazy"> is the set of invertible p-adic integers.

<div class="mw-heading mw-heading2"><h2 id="Basic_properties_of_the_p-adic_gamma_function">Basic properties of the p-adic gamma function</h2></div>

The classical <a href="Gamma_function" title="Gamma function">gamma function</a> satisfies the functional equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (x+1)=x\Gamma (x)}">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (x+1)=x\Gamma (x)}</annotation>

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</math><img src="./2214646d9407377894f7e03405765f87d48e1640.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.615ex; height:2.843ex;" alt="{\displaystyle \Gamma (x+1)=x\Gamma (x)}" loading="lazy"> for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {C} \setminus \mathbb {Z} _{\leq 0}}">

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<annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {C} \setminus \mathbb {Z} _{\leq 0}}</annotation>

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</math><img src="./57801b5fea54b8994eb9180246c0f9e273403110.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.926ex; height:2.843ex;" alt="{\displaystyle x\in \mathbb {C} \setminus \mathbb {Z} _{\leq 0}}" loading="lazy">. This has an analogue with respect to the Morita gamma function:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\Gamma _{p}(x+1)}{\Gamma _{p}(x)}}={\begin{cases}-x,&{\mbox{if }}x\in \mathbb {Z} _{p}^{\times }\\-1,&{\mbox{if }}x\in p\mathbb {Z} _{p}.\end{cases}}}</annotation>

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</math><img src="./51f768df2526af65f0ade51bb83cdeaac475efe7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.726ex; height:6.509ex;" alt="{\displaystyle {\frac {\Gamma _{p}(x+1)}{\Gamma _{p}(x)}}={\begin{cases}-x,&{\mbox{if }}x\in \mathbb {Z} _{p}^{\times }\\-1,&{\mbox{if }}x\in p\mathbb {Z} _{p}.\end{cases}}}" loading="lazy"></dd></dl>

The <a href="Reflection_formula" title="Reflection formula">Euler's reflection formula</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (x)\Gamma (1-x)={\frac {\pi }{\sin {(\pi x)}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (x)\Gamma (1-x)={\frac {\pi }{\sin {(\pi x)}}}}</annotation>

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</math><img src="./0fc31554ced8043e6d9fb8673dde04163134d5c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.835ex; height:5.509ex;" alt="{\displaystyle \Gamma (x)\Gamma (1-x)={\frac {\pi }{\sin {(\pi x)}}}}" loading="lazy"> has its following simple counterpart in the p-adic case:

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{p}(x)\Gamma _{p}(1-x)=(-1)^{x_{0}},}</annotation>

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</math><img src="./4cd1b0b6f89a4b680a1fe28c2c4c5cddaccb24b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.834ex; height:3.009ex;" alt="{\displaystyle \Gamma _{p}(x)\Gamma _{p}(1-x)=(-1)^{x_{0}},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

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</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> is the first digit in the p-adic expansion of x, unless <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in p\mathbb {Z} _{p}}">

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<annotation encoding="application/x-tex">{\displaystyle x\in p\mathbb {Z} _{p}}</annotation>

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</math><img src="./0cf23a28a7ebdf70b09bf3eddd73f1108c9afb1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.949ex; height:2.843ex;" alt="{\displaystyle x\in p\mathbb {Z} _{p}}" loading="lazy">, in which case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=p}">

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<annotation encoding="application/x-tex">{\displaystyle x_{0}=p}</annotation>

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</math><img src="./fb798f9884ccc1a15346d53a3ce34f77c54065b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.652ex; height:2.009ex;" alt="{\displaystyle x_{0}=p}" loading="lazy"> rather than 0.

<div class="mw-heading mw-heading2"><h2 id="Special_values">Special values</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{p}(0)=1,}</annotation>

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</math><img src="./c5defe6f940d6ccb61409cabc69c921fe1865268.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.391ex; height:3.009ex;" alt="{\displaystyle \Gamma _{p}(0)=1,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{p}(1)=-1,}</annotation>

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</math><img src="./2a0ae96173a90566c89c9429fcc6a93fbcb44091.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.2ex; height:3.009ex;" alt="{\displaystyle \Gamma _{p}(1)=-1,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{p}(2)=1,}</annotation>

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</math><img src="./8657afc1c83a1cbdb029b5142841edefd34ed002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.391ex; height:3.009ex;" alt="{\displaystyle \Gamma _{p}(2)=1,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{p}(3)=-2,}</annotation>

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</math><img src="./53bf4d5af8184118bc6932fe50e3bc8f5d537400.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.2ex; height:3.009ex;" alt="{\displaystyle \Gamma _{p}(3)=-2,}" loading="lazy"></dd></dl>

and, in general,

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{p}(n+1)={\frac {(-1)^{n+1}n!}{[n/p]!p^{[n/p]}}}\quad (n\geq 2).}</annotation>

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</math><img src="./8ad37519263a5eb7551eb97d983da03d4df9bfa4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:34.706ex; height:6.843ex;" alt="{\displaystyle \Gamma _{p}(n+1)={\frac {(-1)^{n+1}n!}{[n/p]!p^{[n/p]}}}\quad (n\geq 2).}" loading="lazy"></dd></dl>

At <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\frac {1}{2}}}">

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</math><img src="./2823014b08c738d0755d2397dc6d997a7e59c6c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.427ex; height:5.176ex;" alt="{\displaystyle x={\frac {1}{2}}}" loading="lazy"> the Morita gamma function is related to the <a href="Legendre_symbol" title="Legendre symbol">Legendre symbol</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {a}{p}}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{p}\left({\frac {1}{2}}\right)^{2}=-\left({\frac {-1}{p}}\right).}</annotation>

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</math><img src="./7304d4261e24b9c2627f3fb68b0415fce1f6decb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.541ex; height:6.509ex;" alt="{\displaystyle \Gamma _{p}\left({\frac {1}{2}}\right)^{2}=-\left({\frac {-1}{p}}\right).}" loading="lazy"></dd></dl>

It can also be seen, that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{p}(p^{n})\equiv 1{\pmod {p^{n}}},}">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{p}(p^{n})\equiv 1{\pmod {p^{n}}},}</annotation>

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</math><img src="./af07d11654432cd714248d4a5fcd2fa7a7c1909f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.689ex; height:3.009ex;" alt="{\displaystyle \Gamma _{p}(p^{n})\equiv 1{\pmod {p^{n}}},}" loading="lazy"> hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{p}(p^{n})\to 1}">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{p}(p^{n})\to 1}</annotation>

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</math><img src="./fda1e5eb7b7ed46d2753cfa48c0b5c58f58106e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.486ex; height:3.009ex;" alt="{\displaystyle \Gamma _{p}(p^{n})\to 1}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to \infty }">

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<annotation encoding="application/x-tex">{\displaystyle n\to \infty }</annotation>

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</math><img src="./a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\to \infty }" loading="lazy">.<a href="#cite_note-RobertBook-1">[1]</a>: 369

Other interesting special values come from the <a href="Gross%E2%80%93Koblitz_formula" title="Gross–Koblitz formula">Gross–Koblitz formula</a>, which was first proved by <a href="Cohomology" title="Cohomology">cohomological</a> tools, and later was proved using more elementary methods.<a href="#cite_note-Robert-2">[2]</a> For example,

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{5}\left({\frac {1}{4}}\right)^{2}=-2+{\sqrt {-1}},}</annotation>

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</math><img src="./a18f4273c26aaf3054c0f2e51ca994c9736dcd36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.444ex; height:6.509ex;" alt="{\displaystyle \Gamma _{5}\left({\frac {1}{4}}\right)^{2}=-2+{\sqrt {-1}},}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma _{7}\left({\frac {1}{3}}\right)^{3}={\frac {1-3{\sqrt {-3}}}{2}},}</annotation>

</semantics>

</math><img src="./5f36d95220215b9818d666870e992e9c48b8c32c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.634ex; height:6.509ex;" alt="{\displaystyle \Gamma _{7}\left({\frac {1}{3}}\right)^{3}={\frac {1-3{\sqrt {-3}}}{2}},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-1}}\in \mathbb {Z} _{5}}">

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {-1}}\in \mathbb {Z} _{5}}</annotation>

</semantics>

</math><img src="./ea312a156c1804abd9424f13f142d140682bcefe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.352ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-1}}\in \mathbb {Z} _{5}}" loading="lazy"> denotes the square root with first digit 3, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-3}}\in \mathbb {Z} _{7}}">

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {-3}}\in \mathbb {Z} _{7}}</annotation>

</semantics>

</math><img src="./5875c229517167c965fe9a22adfc2310eb7d2c7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.352ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-3}}\in \mathbb {Z} _{7}}" loading="lazy"> denotes the square root with first digit 2. (Such specifications must always be done if we talk about roots.)

Another example is

<msub>

</mrow>

</msub>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma _{3}\left({\frac {1}{8}}\right)\Gamma _{3}\left({\frac {3}{8}}\right)=-(1+{\sqrt {-2}}),}</annotation>

</semantics>

</math><img src="./f719e8703b5a7a0c5f0a5d5fc11514ff91585daf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.287ex; height:6.176ex;" alt="{\displaystyle \Gamma _{3}\left({\frac {1}{8}}\right)\Gamma _{3}\left({\frac {3}{8}}\right)=-(1+{\sqrt {-2}}),}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-2}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {-2}}}</annotation>

</semantics>

</math><img src="./07901d4c5170c025b49f3b00aa50632e7414d8c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-2}}}" loading="lazy"> is the square root of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -2}</annotation>

</semantics>

</math><img src="./46e5b5b462e546b1d3d7e5f9a23efece405b2e78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -2}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{3}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{3}}</annotation>

</semantics>

</math><img src="./f4e7067a273df719537aeb19b4e17f7f35f571d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.862ex; height:2.676ex;" alt="{\displaystyle \mathbb {Q} _{3}}" loading="lazy"> <a href="Modular_arithmetic" title="Modular arithmetic">congruent</a> to 1 modulo 3.<a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="p-adic_Raabe_formula">p-adic Raabe formula</h2></div>

The Raabe-formula for the classical <a href="Gamma_function" title="Gamma function">Gamma function</a> says that

</mrow>

</mrow>

</msubsup>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{1}\log \Gamma (x+t)dt={\frac {1}{2}}\log(2\pi )+x\log x-x.}</annotation>

</semantics>

</math><img src="./41837a23d5615eb0861ef99a9ad9e8cb41fb93f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.414ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{1}\log \Gamma (x+t)dt={\frac {1}{2}}\log(2\pi )+x\log x-x.}" loading="lazy"></dd></dl>

This has an analogue for the <a href="P-adic_exponential_function#p-adic_logarithm_function" title="P-adic exponential function">Iwasawa logarithm</a> of the Morita gamma function:<a href="#cite_note-4">[4]</a>

<msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {Z} _{p}}\log \Gamma _{p}(x+t)dt=(x-1)(\log \Gamma _{p})'(x)-x+\left\lceil {\frac {x}{p}}\right\rceil \quad (x\in \mathbb {Z} _{p}).}</annotation>

</semantics>

</math><img src="./9564d9091c26e496eb128c17a82bd6b346fb4763.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:64.165ex; height:6.509ex;" alt="{\displaystyle \int _{\mathbb {Z} _{p}}\log \Gamma _{p}(x+t)dt=(x-1)(\log \Gamma _{p})'(x)-x+\left\lceil {\frac {x}{p}}\right\rceil \quad (x\in \mathbb {Z} _{p}).}" loading="lazy"></dd></dl>

The <a href="Ceiling_function" class="mw-redirect" title="Ceiling function">ceiling function</a> to be understood as the p-adic limit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }\left\lceil {\frac {x_{n}}{p}}\right\rceil }">

</mrow>

</munder>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }\left\lceil {\frac {x_{n}}{p}}\right\rceil }</annotation>

</semantics>

</math><img src="./6f5e606e2ad82c0a3253e208778c060c7698f659.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.754ex; height:6.176ex;" alt="{\displaystyle \lim _{n\to \infty }\left\lceil {\frac {x_{n}}{p}}\right\rceil }" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}\to x}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{n}\to x}</annotation>

</semantics>

</math><img src="./bd1fbf73434aa72a428b9b100919b1f6e26f6c84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.492ex; height:2.176ex;" alt="{\displaystyle x_{n}\to x}" loading="lazy"> through rational integers.

<div class="mw-heading mw-heading2"><h2 id="Mahler_expansion">Mahler expansion</h2></div>

The Mahler expansion is similarly important for p-adic functions as the <a href="Taylor_expansion" class="mw-redirect" title="Taylor expansion">Taylor expansion</a> in classical analysis. The Mahler expansion of the p-adic gamma function is the following:<a href="#cite_note-RobertBook-1">[1]</a>: 374

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma _{p}(x+1)=\sum _{k=0}^{\infty }a_{k}{\binom {x}{k}},}</annotation>

</semantics>

</math><img src="./7d00006774693b81fccc9e87b27de67d49aae10f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.211ex; height:7.009ex;" alt="{\displaystyle \Gamma _{p}(x+1)=\sum _{k=0}^{\infty }a_{k}{\binom {x}{k}},}" loading="lazy"></dd></dl>

where the <a href="Sequence" title="Sequence">sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{k}}</annotation>

</semantics>

</math><img src="./05e256a120c3ab9f8958de71acdf81cd75065e3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.319ex; height:2.009ex;" alt="{\displaystyle a_{k}}" loading="lazy"> is defined by the following identity:

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{\infty }(-1)^{k+1}a_{k}{\frac {x^{k}}{k!}}={\frac {1-x^{p}}{1-x}}\exp \left(x+{\frac {x^{p}}{p}}\right).}</annotation>

</semantics>

</math><img src="./b04cbfa1fe6e68db4eaecc376254280debe45f71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.626ex; height:7.009ex;" alt="{\displaystyle \sum _{k=0}^{\infty }(-1)^{k+1}a_{k}{\frac {x^{k}}{k!}}={\frac {1-x^{p}}{1-x}}\exp \left(x+{\frac {x^{p}}{p}}\right).}" loading="lazy"></dd></dl>

<ul><li><a href="Gross%E2%80%93Koblitz_formula" title="Gross–Koblitz formula">Gross–Koblitz formula</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFBoyarsky1980" class="citation cs2">Boyarsky, Maurizio (1980), "p-adic gamma functions and Dwork cohomology", <a href="Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a>, 257 (2): 359–369, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1998301">10.2307/1998301</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9947">0002-9947</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1998301">1998301</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0552263">0552263</a></cite></li>

<li><cite id="CITEREFDiamond1977" class="citation cs2">Diamond, Jack (1977), "The p-adic log gamma function and p-adic Euler constants", <a href="Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a>, 233: 321–337, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1997840">10.2307/1997840</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9947">0002-9947</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1997840">1997840</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0498503">0498503</a></cite></li>

<li><cite id="CITEREFDiamond1984" class="citation cs2">Diamond, Jack (1984), "p-adic gamma functions and their applications", in <a href="Chudnovsky_brothers" title="Chudnovsky brothers">Chudnovsky, David V.</a>; Chudnovsky, Gregory V.; Cohn, Henry; et al. (eds.), Number theory (New York, 1982), Lecture Notes in Math., vol. 1052, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, pp. 168–175, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0071542">10.1007/BFb0071542</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-12909-7</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0750664">0750664</a></cite></li>

<li><cite id="CITEREFDwork1964" class="citation cs2">Dwork, Bernard (1964), "On the zeta function of a hypersurface. II", <a href="Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a>, Second Series, 80 (2): 227–299, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1970392">10.2307/1970392</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1970392">1970392</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0188215">0188215</a></cite></li>

<li><cite id="CITEREFMorita1975" class="citation cs2">Morita, Yasuo (1975), "A p-adic analogue of the Γ-function", Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics, 22 (2): 255–266, <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2261%2F6494">2261/6494</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0040-8980">0040-8980</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0424762">0424762</a></cite></li>

<li><cite id="CITEREFOverholtzer1952" class="citation cs2">Overholtzer, Gordon (1952), "Sum functions in elementary p-adic analysis", <a href="American_Journal_of_Mathematics" title="American Journal of Mathematics">American Journal of Mathematics</a>, 74 (2): 332–346, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2371998">10.2307/2371998</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9327">0002-9327</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2371998">2371998</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0048493">0048493</a></cite></li></ul>

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<cite id="CITEREFRobert2000" class="citation book cs1">Robert, Alain M. (2000). A course in p-adic analysis. New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>.</cite>

</li>

<li id="cite_note-Robert-2"><a href="#cite_ref-Robert_2-0">^</a> <cite id="CITEREFRobert2001" class="citation journal cs1">Robert, Alain M. (2001). <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=RSMUP_2001__105__157_0">"The Gross-Koblitz formula revisited"</a>. Rendiconti del Seminario Matematico della Università di Padova. The Mathematical Journal of the University of Padova. 105: 157–170. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jnt.2009.08.005">10.1016/j.jnt.2009.08.005</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2437%2F90539">2437/90539</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0041-8994">0041-8994</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1834987">1834987</a>.</cite>

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