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The factorial of a rational number, the gamma function not used

  1. Aug 18, 2014 #1
    My first question is: is this formula (at the bottom) a known formula?
    In this subject i haven't explained how i build up the formula.
    So far i think it is equal to the gamma function of Euler with

    [tex] \Gamma\left(\frac{m_1}{m_2}+1\right)= \frac{m_1}{m_2}\ ![/tex]

    with

    [tex] m_1 , m_2 \in \mathbf{N} [/tex]

    and

    [tex] 1<m_2 [/tex]

    This gamma function however I didn't use.
    The next limit i write like L(q,x) in de last formula.

    [tex] L(q,x)=\prod^{\infty}_{k=0}\frac{(k+q+x) (k+1)}{(k+q)(k+1+x)}[/tex]

    with

    [tex] q\neq [/tex]

    0,-1,-2,-3, …… and

    [tex] x\neq [/tex]

    –1,-2,-3,-4, ……..

    This is the formula where i use the factorial symbol ! because i think that it gives no problems with arguments with real values.

    [tex] \frac{ m_1}{m_2}\ !=\left(m_1\ ! \prod^{ m_2-1}_{ i=1}L\left(1+ i\frac{ m_1}{m_2}, \frac{ m_1}{m_2}\right) \right)^\frac{1}{m_2} [/tex]
     
     
    with

    [tex] m_1 , m_2 \in\mathbf{N} [/tex]

    and

    [tex] 1<m_2 [/tex]

    Please feel free to react.
     
  2. jcsd
  3. Aug 19, 2014 #2
    Does it only work for positive natural numbers? Even if limited by those inputs I still like your 'q' and 'x' general form (assuming this identity is correct).
     
  4. Aug 21, 2014 #3
    @mesa,
    The input is a positive rational number m1/m2.
    For an input of a negative rational number the relation between x! and (-x)! according to my definition (which i didn't show here) of the factorial (!) is:

    x!(-x)!L(1-x,x)=1
     
  5. Aug 21, 2014 #4
    You did write that (sorry, was too focused elsewhere).
     
  6. Aug 30, 2014 #5
    I made a mistake in the last equation. It must be:
    x!(-x)!=L(1-x,x)
    This formula compares to Euler's reflection formula.
     
  7. Oct 18, 2014 #6
    This is a factorial summation formula.

    (x+y)!L(x+1,y)=x!y!
    For example if y=0
    (x+0)!L(x+1,0)=x!L(x+1,0)=x!=x!0!
    If y=1
    (x+1)!L(x+1,1)= (x+1)!/(x+1)=x!1!=x!
     
  8. Oct 26, 2014 #7
    The next formula is about binomial coefficients.
    If I use my definition of the factorial in this formula:
    [tex]
    {x \choose y}=\frac{x\ !}{y\ !(x-y)\ !}
    [/tex]
    (x, y are rational numbers) then the next equation appears :
    [tex]
    {x \choose y}L(1+x-y,y)={x \choose y}\prod^{\infty}_{k=0}\frac{(k+x+1) (k+1)}{(k+1+x-y)(k+1+y)}={x \choose y}\prod^{\infty}_{k=1}\frac{(k+x) k}{(k+x-y)(k+y)}=1
    [/tex]
    For example if y=0:
    [tex]
    {x \choose 0}L(1+x,0)={x \choose 0}\prod^{\infty}_{k=0}\frac{(k+x+1) (k+1)}{(k+1+x)(k+1)}=1
    [/tex]
    Or if x=1 and y=1/2
    [tex]
    {1 \choose \frac{1}{2} }L(1+1-\frac{1}{2},\frac{1}{2})={1 \choose \frac{1}{2}}\prod^{\infty}_{k=0}\frac{(k+1+1) (k+1)}{(k+1+1-\frac{1}{2})(k+1+\frac{1}{2})}={1 \choose \frac{1}{2}}\prod^{\infty}_{k=1}\frac{(k+1) k}{(k+1-\frac{1}{2})(k+\frac{1}{2})}={1 \choose \frac{1}{2}}\prod^{\infty}_{k=1}\frac{(k+1) k}{(k+\frac{1}{2})^2}={1 \choose \frac{1}{2}}\frac {\pi}{4}=1
    [/tex]
    Does this give the same results when using the 'gamma function'?
     
  9. Oct 26, 2014 #8
    The remaining text (x=1 and y=1/2):

    [tex]
    ={1 \choose \frac{1}{2} }L(1+1-\frac{1}{2},\frac{1}{2})={1 \choose \frac{1}{2}}\frac {\pi}{4}=1
    [/tex]
     
  10. Feb 24, 2015 #9
    I think I can approach the Euler–Mascheroni constant with the next formula:

    [tex]
    - \gamma =\lim_{n \rightarrow \infty}((\prod^{\infty}_{k=1}(\frac{(k+1)^\frac{1}{n} k^\frac{n-1}{n}}{(k+\frac{1}{n})})-1)n)
    [/tex]
     
  11. Mar 22, 2015 #10
    To complete the formula about the Euler-Mascheroni constant I'll add this one:

    [tex]
    \gamma =
    \lim_{n \rightarrow \infty}
    ((\prod^{\infty}_{k=1}
    (\frac{(k+1)^\frac{-1}{n} k^\frac{n+1}{n}}{(k-\frac{1}{n})})-1)n)
    [/tex]
     
  12. Apr 5, 2015 #11
    I also think the digamma function for
    [tex]
    x\in\mathbb{Q}
    [/tex] can be expressed in the next formula:

    [tex]
    \Psi(x)=-\gamma+\lim_{n\rightarrow\pm\infty}(n(1-L(x,\frac{1}{n}))
    [/tex]
     
  13. Oct 6, 2015 #12
    Of course you have already noticed that a multiplication formula can easely be derivered from the summation formula.


    For [tex] n \in \mathbf{N}, 1<n [/tex] and [tex] x \in \mathbf{Q} [/tex]

    [tex]
    (nx) ! = \frac {{x !}^n}{\prod\limits_{k=1}^{n-1}L\left(1+ kx,x\right)}
    [/tex]
     
  14. Oct 20, 2015 #13
    I think Leo Pochhammer did a good thing by changing the factorial function from a unary operation into a binary operation.

    As I see it this is the definition of the rising factorial:



    [tex]

    n, m \in \mathbf{N} \ (starting\ with\ zero),\ x \in \mathbf{C}

    [/tex]


    [tex]
    X1: \ Pochhammer(x,0)=1 \ (definition\ one)\\
    [/tex]
    [tex]
    X2: \ Pochhammer(x,n+1)=Pochhammer(x,n)*(x+n)\ (definition\ two)
    [/tex]



    My idea is to use the symbol ! as a binary operator like this:



    [tex]
    X1: \ x!0=1\ (definition\ one)\\
    [/tex]
    [tex]
    X2: \ x!(n+1)=x!n*(x+n)\ (definition\ two)
    [/tex]


    Now a theorem can be proved bij these definitions and mathematical induction.



    [tex] Pochhammer(x,n+m)=Pochhammer(x,n)*Pochhammer(x+n,m)

    [/tex]


    This is my formula, I prefer to use [tex]x!y[/tex] instead of [tex]Pochhammer(x,y)[/tex]:



    [tex]

    for\ x,y \in \mathbf{Q}

    [/tex]



    [tex]

    x!y=Pochhammer(x,y)=\frac{1!y}{L(x,y)}

    [/tex]
     
  15. Apr 22, 2016 #14
    My next formula is also about the digamma function or more precisely the derivative of the natural logarithm of the factorial function as I defined this, the input x shift by 1. I use the notation of the digamma function [itex] \Psi(x)[/itex] because I think, and maybe somebody can prove this, it is the same as my formula.





    [tex] \Psi(x)=\lim_{h\rightarrow 0}\ln{h!^\frac{1}{h}}-\lim_{h\rightarrow 0}\ln{L(x,h)^\frac{1}{h}}=-\gamma-\lim_{h\rightarrow 0}\ln{L(x,h)^\frac{1}{h}} [/tex]
     
  16. May 10, 2016 #15
    I Want to make a few more steps. Hopefully mathematically correct.





    [tex]



    \lim_{h\rightarrow 0}\ln{h!^\frac{1}{h}}=\lim_{h\rightarrow 0}\frac{\ln{(0+h)!}-\ln{0!}}{h}=\Psi(1)=-\gamma



    [/tex]





    and



    [tex]



    \lim_{h\rightarrow 0}\ln{L(x,h)^\frac{1}{h}}=

    \lim_{h\rightarrow 0}\ln\prod^{\infty}_{k=0}{\left(\frac{(k+x+h) (k+1)}{(k+x)(k+1+h)}\right)^\frac{1}{h}}=\lim_{h\rightarrow 0}\sum^{\infty}_{k=0}{\ln\left(\frac{(k+x+h) (k+1)}{(k+x)(k+1+h)}\right)^\frac{1}{h}}=\lim_{h\rightarrow 0}\sum^{\infty}_{k=0}{\frac{\ln(k+x+h)-\ln(k+x)}{h}-\frac{\ln(k+1+h)-\ln(k+1)}{h}}=\sum^{\infty}_{k=0}{\lim_{h\rightarrow 0}\frac{\ln(k+x+h)-\ln(k+x)}{h}-\frac{\ln(k+1+h)-\ln(k+1)}{h}}=\sum^{\infty}_{k=0}{\frac{1}{k+x}-\frac{1}{k+1}}



    [/tex]
     
  17. May 16, 2016 #16
    I started with a definition of the factorial function for x is a rational number. Then I could derive a formula that gives the derivative of the natural logarithm of the factorial function (shifted by one this match with the digamma function). I think and tell me if I am wrong, it is easy to derive 'the Weierstrass representation‏ of the gamma function' by integrating this function and so on.



    [tex]

    \Psi(x)=-\gamma +\sum^{\infty}_{k=0}{\left(\frac{1}{k+1}-\frac{1}{k+x}\right)}

    \Rightarrow

    [/tex]



    [tex]

    \ln\Gamma(x)=-\gamma x+\sum^{\infty}_{k=0}{\left(\frac{x-1}{k+1}-\ln\frac{k+x}{k+1}\right)}+c

    \Rightarrow

    [/tex]



    [tex]

    \Gamma(x)=e^{-\gamma x}e^{\sum\limits^{\infty}_{k=0}{\left(\frac{x-1}{k+1}-\ln\frac{k+x}{k+1}\right)}}e^c=

    [/tex]



    [tex]
    e^{-\gamma x+c}\prod^{\infty}_{k=0}{e^{\left(\frac{x-1}{k+1}\right)}\left(\frac{k+1}{k+x}\right)}=

    [/tex]



    [tex]

    e^{-\gamma x+c}\prod^{\infty}_{k=1}{e^{\left(\frac{x-1}{k}\right)}\left(\frac{k}{k+x-1}\right)}=

    [/tex]



    According to the definition



    [tex]

    \Gamma(1)=1\\

    c=\gamma\\

    \Gamma(x)=e^{-\gamma x+\gamma}\prod^{\infty}_{k=1}{e^{\frac{x-1}{k}}\frac{k}{k+x-1}}

    [/tex]



    [tex]

    \Gamma(x+1)=

    e^{-\gamma x}\prod^{\infty}_{k=1}{e^{\frac{x}{k}}\frac{k}{k+x}}

    [/tex]



    According to the definition



    [tex]

    \Gamma(x+1)=\Gamma(x)x\\

    \Gamma(x)=\frac{e^{-\gamma x}}{x}\prod^{\infty}_{k=1}{e^{\frac{x}{k}}\frac{k}{k+x}}

    [/tex]



    [tex]

    \Gamma(x)=\frac{e^{-\gamma x}}{x}\prod^{\infty}_{k=1}{e^{\frac{x}{k}}(1+\frac{x}{k})^{-1}}

    [/tex]
     
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