p-adic gamma function

<h2 id="definition">Definition</h2>
The p-adic gamma function is the unique continuous function of a <a href="/facts/P-adic_integer/SDgDbkLF">p-adic integer</a> x (with values in 
 
 
 
 
 
 Z
 
 
 p
 
 
 
 
 {\displaystyle \mathbb {Z} _{p}}
 
) such that

Γ
 
 p
 
 
 (
 x
 )
 =
 (
 −
 1
 
 )
 
 x
 
 
 
 ∏
 
 0
 <
 i
 <
 x
 ,
  
 p
 
 ∤
 
 i
 
 
 i
 
 
 {\displaystyle \Gamma _{p}(x)=(-1)^{x}\prod _{0<i<x,\ p\,\nmid \,i}i}

for <a href="/facts/Positive_integer/u65og8JE">positive integers</a> x, where the product is restricted to integers i not <a href="/facts/Divisibility/DKNa2GvC">divisible</a> by p. As the positive integers are dense with respect to the p-adic topology in 
 
 
 
 
 
 Z
 
 
 p
 
 
 
 
 {\displaystyle \mathbb {Z} _{p}}
 
, 
 
 
 
 
 Γ
 
 p
 
 
 (
 x
 )
 
 
 {\displaystyle \Gamma _{p}(x)}
 
 can be extended uniquely to the whole of 
 
 
 
 
 
 Z
 
 
 p
 
 
 
 
 {\displaystyle \mathbb {Z} _{p}}
 
. Here 
 
 
 
 
 
 Z
 
 
 p
 
 
 
 
 {\displaystyle \mathbb {Z} _{p}}
 
 is the ring of <a href="/facts/P-adic_integer/SDgDbkLF">p-adic integers</a>. It follows from the definition that the values of 
 
 
 
 
 Γ
 
 p
 
 
 (
 
 Z
 
 )
 
 
 {\displaystyle \Gamma _{p}(\mathbb {Z} )}
 
 are invertible in 
 
 
 
 
 
 Z
 
 
 p
 
 
 
 
 {\displaystyle \mathbb {Z} _{p}}
 
; this is because these values are products of integers not divisible by p, and this property holds after the continuous extension to 
 
 
 
 
 
 Z
 
 
 p
 
 
 
 
 {\displaystyle \mathbb {Z} _{p}}
 
. Thus 
 
 
 
 
 Γ
 
 p
 
 
 :
 
 
 Z
 
 
 p
 
 
 →
 
 
 Z
 
 
 p
 
 
 ×
 
 
 
 
 {\displaystyle \Gamma _{p}:\mathbb {Z} _{p}\to \mathbb {Z} _{p}^{\times }}
 
. Here 
 
 
 
 
 
 Z
 
 
 p
 
 
 ×
 
 
 
 
 {\displaystyle \mathbb {Z} _{p}^{\times }}
 
 is the set of invertible p-adic integers.

<h2 id="basic-properties-of-the-p-adic-gamma-function">Basic properties of the p-adic gamma function</h2>
The classical <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a> satisfies the functional equation 
 
 
 
 Γ
 (
 x
 +
 1
 )
 =
 x
 Γ
 (
 x
 )
 
 
 {\displaystyle \Gamma (x+1)=x\Gamma (x)}
 
 for any 
 
 
 
 x
 ∈
 
 C
 
 ∖
 
 
 Z
 
 
 ≤
 0
 
 
 
 
 {\displaystyle x\in \mathbb {C} \setminus \mathbb {Z} _{\leq 0}}
 
. This has an analogue with respect to the Morita gamma function:

Γ
                
                  p
                
              
              (
              x
              +
              1
              )
            
            
              
                Γ
                
                  p
                
              
              (
              x
              )
            
          
        
        =
        
          
            {
            
              
                
                  −
                  x
                  ,
                
                
                  
                    
                      if 
                    
                  
                  x
                  ∈
                  
                    
                      Z
                    
                    
                      p
                    
                    
                      ×
                    
                  
                
              
              
                
                  −
                  1
                  ,
                
                
                  
                    
                      if 
                    
                  
                  x
                  ∈
                  p
                  
                    
                      Z
                    
                    
                      p
                    
                  
                  .
                
              
            
            
          
        
      
    
    {\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}}}

The <a href="/facts/Reflection_formula/aKUhTbH0">Euler's reflection formula</a> 
 
 
 
 Γ
 (
 x
 )
 Γ
 (
 1
 −
 x
 )
 =
 
 
 π
 
 sin
 ⁡
 
 (
 π
 x
 )
 
 
 
 
 
 
 {\displaystyle \Gamma (x)\Gamma (1-x)={\frac {\pi }{\sin {(\pi x)}}}}
 
 has its following simple counterpart in the p-adic case:

Γ
          
            p
          
        
        (
        x
        )
        
          Γ
          
            p
          
        
        (
        1
        −
        x
        )
        =
        (
        −
        1
        
          )
          
            
              x
              
                0
              
            
          
        
        ,
      
    
    {\displaystyle \Gamma _{p}(x)\Gamma _{p}(1-x)=(-1)^{x_{0}},}

where 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 is the first digit in the p-adic expansion of x, unless 
 
 
 
 x
 ∈
 p
 
 
 Z
 
 
 p
 
 
 
 
 {\displaystyle x\in p\mathbb {Z} _{p}}
 
, in which case 
 
 
 
 
 x
 
 0
 
 
 =
 p
 
 
 {\displaystyle x_{0}=p}
 
 rather than 0.

<h2 id="special-values">Special values</h2>

Γ
          
            p
          
        
        (
        0
        )
        =
        1
        ,
      
    
    {\displaystyle \Gamma _{p}(0)=1,}

Γ
          
            p
          
        
        (
        1
        )
        =
        −
        1
        ,
      
    
    {\displaystyle \Gamma _{p}(1)=-1,}

Γ
          
            p
          
        
        (
        2
        )
        =
        1
        ,
      
    
    {\displaystyle \Gamma _{p}(2)=1,}

Γ
          
            p
          
        
        (
        3
        )
        =
        −
        2
        ,
      
    
    {\displaystyle \Gamma _{p}(3)=-2,}

and, in general,

Γ
          
            p
          
        
        (
        n
        +
        1
        )
        =
        
          
            
              (
              −
              1
              
                )
                
                  n
                  +
                  1
                
              
              n
              !
            
            
              [
              n
              
                /
              
              p
              ]
              !
              
                p
                
                  [
                  n
                  
                    /
                  
                  p
                  ]
                
              
            
          
        
        
        (
        n
        ≥
        2
        )
        .
      
    
    {\displaystyle \Gamma _{p}(n+1)={\frac {(-1)^{n+1}n!}{[n/p]!p^{[n/p]}}}\quad (n\geq 2).}

At 
 
 
 
 x
 =
 
 
 1
 2
 
 
 
 
 {\displaystyle x={\frac {1}{2}}}
 
 the Morita gamma function is related to the <a href="/facts/Legendre_symbol/MwHrJ7as">Legendre symbol</a> 
 
 
 
 
 (
 
 
 a
 p
 
 
 )
 
 
 
 {\displaystyle \left({\frac {a}{p}}\right)}
 
:

Γ
          
            p
          
        
        
          
            (
            
              
                1
                2
              
            
            )
          
          
            2
          
        
        =
        −
        
          (
          
            
              
                −
                1
              
              p
            
          
          )
        
        .
      
    
    {\displaystyle \Gamma _{p}\left({\frac {1}{2}}\right)^{2}=-\left({\frac {-1}{p}}\right).}

It can also be seen, that 
 
 
 
 
 Γ
 
 p
 
 
 (
 
 p
 
 n
 
 
 )
 ≡
 1
 
 
 (
 mod
 
 
 p
 
 n
 
 
 )
 
 ,
 
 
 {\displaystyle \Gamma _{p}(p^{n})\equiv 1{\pmod {p^{n}}},}
 
 hence 
 
 
 
 
 Γ
 
 p
 
 
 (
 
 p
 
 n
 
 
 )
 →
 1
 
 
 {\displaystyle \Gamma _{p}(p^{n})\to 1}
 
 as 
 
 
 
 n
 →
 ∞
 
 
 {\displaystyle n\to \infty }
 
.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>: 369 
Other interesting special values come from the <a href="/facts/Gross%25E2%2580%2593Koblitz_formula/XR5pdp80">Gross–Koblitz formula</a>, which was first proved by <a href="/facts/Cohomology/J7FJzTNh">cohomological</a> tools, and later was proved using more elementary methods.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> For example,

Γ
          
            5
          
        
        
          
            (
            
              
                1
                4
              
            
            )
          
          
            2
          
        
        =
        −
        2
        +
        
          
            −
            1
          
        
        ,
      
    
    {\displaystyle \Gamma _{5}\left({\frac {1}{4}}\right)^{2}=-2+{\sqrt {-1}},}

Γ
          
            7
          
        
        
          
            (
            
              
                1
                3
              
            
            )
          
          
            3
          
        
        =
        
          
            
              1
              −
              3
              
                
                  −
                  3
                
              
            
            2
          
        
        ,
      
    
    {\displaystyle \Gamma _{7}\left({\frac {1}{3}}\right)^{3}={\frac {1-3{\sqrt {-3}}}{2}},}

where 
 
 
 
 
 
 −
 1
 
 
 ∈
 
 
 Z
 
 
 5
 
 
 
 
 {\displaystyle {\sqrt {-1}}\in \mathbb {Z} _{5}}
 
 denotes the square root with first digit 3, and 
 
 
 
 
 
 −
 3
 
 
 ∈
 
 
 Z
 
 
 7
 
 
 
 
 {\displaystyle {\sqrt {-3}}\in \mathbb {Z} _{7}}
 
 denotes the square root with first digit 2. (Such specifications must always be done if we talk about roots.)
Another example is

Γ
          
            3
          
        
        
          (
          
            
              1
              8
            
          
          )
        
        
          Γ
          
            3
          
        
        
          (
          
            
              3
              8
            
          
          )
        
        =
        −
        (
        1
        +
        
          
            −
            2
          
        
        )
        ,
      
    
    {\displaystyle \Gamma _{3}\left({\frac {1}{8}}\right)\Gamma _{3}\left({\frac {3}{8}}\right)=-(1+{\sqrt {-2}}),}

where 
 
 
 
 
 
 −
 2
 
 
 
 
 {\displaystyle {\sqrt {-2}}}
 
 is the square root of 
 
 
 
 −
 2
 
 
 {\displaystyle -2}
 
 in 
 
 
 
 
 
 Q
 
 
 3
 
 
 
 
 {\displaystyle \mathbb {Q} _{3}}
 
 <a href="/facts/Modular_arithmetic/0wtRa5dU">congruent</a> to 1 modulo 3.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<h2 id="p-adic-raabe-formula">p-adic Raabe formula</h2>
The Raabe-formula for the classical <a href="/facts/Gamma_function/SjGQcnyw">Gamma function</a> says that

∫
          
            0
          
          
            1
          
        
        log
        ⁡
        Γ
        (
        x
        +
        t
        )
        d
        t
        =
        
          
            1
            2
          
        
        log
        ⁡
        (
        2
        π
        )
        +
        x
        log
        ⁡
        x
        −
        x
        .
      
    
    {\displaystyle \int _{0}^{1}\log \Gamma (x+t)dt={\frac {1}{2}}\log(2\pi )+x\log x-x.}

This has an analogue for the <a href="/facts/P-adic_exponential_function/T5wK0u3g">Iwasawa logarithm</a> of the Morita gamma function:<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

∫
          
            
              
                Z
              
              
                p
              
            
          
        
        log
        ⁡
        
          Γ
          
            p
          
        
        (
        x
        +
        t
        )
        d
        t
        =
        (
        x
        −
        1
        )
        (
        log
        ⁡
        
          Γ
          
            p
          
        
        
          )
          ′
        
        (
        x
        )
        −
        x
        +
        
          ⌈
          
            
              x
              p
            
          
          ⌉
        
        
        (
        x
        ∈
        
          
            Z
          
          
            p
          
        
        )
        .
      
    
    {\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}).}

The <a href="/facts/Ceiling_function/ssehyMMf">ceiling function</a> to be understood as the p-adic limit 
 
 
 
 
 lim
 
 n
 →
 ∞
 
 
 
 ⌈
 
 
 
 x
 
 n
 
 
 p
 
 
 ⌉
 
 
 
 {\displaystyle \lim _{n\to \infty }\left\lceil {\frac {x_{n}}{p}}\right\rceil }
 
 such that 
 
 
 
 
 x
 
 n
 
 
 →
 x
 
 
 {\displaystyle x_{n}\to x}
 
 through rational integers.

<h2 id="mahler-expansion">Mahler expansion</h2>
The Mahler expansion is similarly important for p-adic functions as the <a href="/facts/Taylor_expansion/M0aTuftH">Taylor expansion</a> in classical analysis. The Mahler expansion of the p-adic gamma function is the following:<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>: 374

Γ
          
            p
          
        
        (
        x
        +
        1
        )
        =
        
          ∑
          
            k
            =
            0
          
          
            ∞
          
        
        
          a
          
            k
          
        
        
          
            
              (
            
            
              x
              k
            
            
              )
            
          
        
        ,
      
    
    {\displaystyle \Gamma _{p}(x+1)=\sum _{k=0}^{\infty }a_{k}{\binom {x}{k}},}

where the <a href="/facts/Sequence/gV2S4uqp">sequence</a> 
 
 
 
 
 a
 
 k
 
 
 
 
 {\displaystyle a_{k}}
 
 is defined by the following identity:

∑
          
            k
            =
            0
          
          
            ∞
          
        
        (
        −
        1
        
          )
          
            k
            +
            1
          
        
        
          a
          
            k
          
        
        
          
            
              x
              
                k
              
            
            
              k
              !
            
          
        
        =
        
          
            
              1
              −
              
                x
                
                  p
                
              
            
            
              1
              −
              x
            
          
        
        exp
        ⁡
        
          (
          
            x
            +
            
              
                
                  x
                  
                    p
                  
                
                p
              
            
          
          )
        
        .
      
    
    {\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).}

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Gross%25E2%2580%2593Koblitz_formula/XR5pdp80">Gross–Koblitz formula</a></li></ul>

<ul><li>Boyarsky, Maurizio (1980), "p-adic gamma functions and Dwork cohomology", <a href="/facts/Transactions_of_the_American_Mathematical_Society/JmKtyCwK">Transactions of the American Mathematical Society</a>, 257 (2): 359–369, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1998301">10.2307/1998301</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0002-9947">0002-9947</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1998301">1998301</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0552263">0552263</a></li>
<li>Diamond, Jack (1977), "The p-adic log gamma function and p-adic Euler constants", <a href="/facts/Transactions_of_the_American_Mathematical_Society/JmKtyCwK">Transactions of the American Mathematical Society</a>, 233: 321–337, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1997840">10.2307/1997840</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0002-9947">0002-9947</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1997840">1997840</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0498503">0498503</a></li>
<li>Diamond, Jack (1984), "p-adic gamma functions and their applications", in <a href="/facts/Chudnovsky_brothers/HXIbzEXL">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="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, pp. 168–175, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBFb0071542">10.1007/BFb0071542</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-12909-7, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0750664">0750664</a></li>
<li>Dwork, Bernard (1964), "On the zeta function of a hypersurface. II", <a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a>, Second Series, 80 (2): 227–299, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1970392">10.2307/1970392</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1970392">1970392</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0188215">0188215</a></li>
<li>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="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/2261%2F6494">2261/6494</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0040-8980">0040-8980</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0424762">0424762</a></li>
<li>Overholtzer, Gordon (1952), "Sum functions in elementary p-adic analysis", <a href="/facts/American_Journal_of_Mathematics/lkgftcD7">American Journal of Mathematics</a>, 74 (2): 332–346, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2371998">10.2307/2371998</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0002-9327">0002-9327</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2371998">2371998</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0048493">0048493</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Robert, Alain M. (2000). A course in p-adic analysis. New York: Springer-Verlag. <a href="/wiki/Springer-Verlag" target="_blank">/wiki/Springer-Verlag</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Robert, Alain M. (2001). "The Gross-Koblitz formula revisited". Rendiconti del Seminario Matematico della Università di Padova. The Mathematical Journal of the University of Padova. 105: 157–170. doi:10.1016/j.jnt.2009.08.005. hdl:2437/90539. ISSN 0041-8994. MR 1834987. <a href="http://www.numdam.org/item?id=RSMUP_2001__105__157_0" target="_blank">http://www.numdam.org/item?id=RSMUP_2001__105__157_0</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Cohen, H. (2007). Number Theory. Vol. 2. New York: Springer Science+Business Media. p. 406. <a href="/wiki/Springer_Science%2BBusiness_Media" target="_blank">/wiki/Springer_Science%2BBusiness_Media</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Cohen, Henri; Eduardo, Friedman (2008). "Raabe's formula for p-adic gamma and zeta functions". Annales de l'Institut Fourier. 88 (1): 363–376. doi:10.5802/aif.2353. hdl:10533/139530. MR 2401225. <a href="http://www.numdam.org/item/AIF_2008__58_1_363_0/" target="_blank">http://www.numdam.org/item/AIF_2008__58_1_363_0/</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Robert, Alain M. (2000). A course in p-adic analysis. New York: Springer-Verlag. <a href="/wiki/Springer-Verlag" target="_blank">/wiki/Springer-Verlag</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
</ol>

p-adic gamma function open-in-new

p-adic gamma function