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Find formula of a Gamma[] product with complex conjugate pair of numbers?

GROUPS:

Wolfram Language knows about some simplifications for the product of the Gammafunction for complex conjugate numbers, i.e.

In[1]:= Gamma[n I] Gamma[-n I] // FullSimplify

Out[1]= (\[Pi] Csch[n \[Pi]])/n

In[2]:= Gamma[1 + n I] Gamma[1 - n I] // FullSimplify

Out[2]= n \[Pi] Csch[n \[Pi]]


In[3]:= Gamma[2 + n I] Gamma[2 - n I] // FullSimplify

Out[3]= n (1 + n^2) \[Pi] Csch[n \[Pi]]

In[4]:= Gamma[3 + n I] Gamma[3 - n I] // FullSimplify

Out[4]= n (4 + 5 n^2 + n^4) \[Pi] Csch[n \[Pi]]

In[5]:= Gamma[4 + n I] Gamma[4 - n I] // FullSimplify

Out[5]= Gamma[4 - I n] Gamma[4 + I n]

... but at some point it is stuck. With the help of WL I found these Identities for m=4,5 and 6.

In[8]:= N[Table[{Gamma[4 + n I] Gamma[
       4 - n I] == (\[Pi] n (n^6 + 14 n^4 + 49 n^2 + 36))/
     Sinh[n \[Pi]]}, {n, 1, 4}], 20]

Out[8]= {{True}, {True}, {True}, {True}}
In[9]:= N[
 Table[{Gamma[5 + n I] Gamma[
      5 - n I] == (\[Pi] n (n^8 + 30 n^6 + 273 n^4 + 820 n^2 + 576))/
    Sinh[n \[Pi]]}, {n, 1, 4}], 20]

Out[9]= {{True}, {True}, {True}, {True}}
In[10]:= N[
 Table[{Gamma[6 + n I] Gamma[
      6 - n I] == (\[Pi] n (n^10 + 55 n^8 + 1023 n^6 + 7645 n^4 + 
       21076 n^2 + 14400))/Sinh[n \[Pi]]}, {n, 1, 4}], 20]

Out[10]= {{True}, {True}, {True}, {True}}

And of course more identities for other natural numbers can be found with some effort. Maybe someone can find a closed formula. for the case Gamma[m+I n]Gamma[m-I n], where m is a natural number.

POSTED BY: Oliver Seipel
Answer
1 month ago

Hi

It's a simple formula because I found it in a few minutes.

HoldForm[Gamma[m + n I] Gamma[m - n I] == n π Csch[n π]*Product[(-1)^(2 j) (-n*I + j) (n*I + j ), {j, 1, m - 1}] == 
n π Csch[n π]*Pochhammer[1 - I n, -1 + m] Pochhammer[1 + I n, -1 + m]] // TeXForm

$$\Gamma (m+n i) \Gamma (m-n i)=n \pi \text{csch}(n \pi ) \prod _{j=1}^{m-1} (-1)^{2 j} (-n i+j) (n i+j)=n \pi \text{csch}(n \pi ) (1-i n)_{-1+m} (1+i n)_{-1+m}$$

 f[m_] := n π Csch[n π]*Product[(-1)^(2 j) (-n*I + j) (n*I + j ), {j, 1, m - 1}]
 f[5] // FullSimplify // Expand
 (*576 n π Csch[n π] + 820 n^3 π Csch[n π] + 273 n^5 π Csch[n π] + 30 n^7 π Csch[n π] + n^9 π Csch[n π] *)

 g[m_] := Gamma[m + n I] Gamma[m - n I]
 g[5] // FunctionExpand // Expand
 (*576 n π Csch[n π] + 820 n^3 π Csch[n π] + 273 n^5 π Csch[n π] + 30 n^7 π Csch[n π] + n^9 π Csch[n π] *)

Regards,MI

POSTED BY: Mariusz Iwaniuk
Answer
1 month ago

Hi MI, nice! Regards, OS

POSTED BY: Oliver Seipel
Answer
1 month ago

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