Why are the hypergeometric functions called “hypergeometric”?












4












$begingroup$


I was wondering where the term "hypergeometric" for the hypergeometric function $_2F_1(a, b; c; z)$ comes from. Wikipedia says that the term was coined by J. Wallis, but I couldn't find any (mathematical) reason why these functions are anything like hyper-geometric.



Does anyone know where this comes from?






terminology special-functions







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    mathworld.wolfram.com/HypergeometricFunction.html
    $endgroup$
    – Karn Watcharasupat
    Apr 12 '18 at 8:42






  • 1




    $begingroup$
    see also math.stackexchange.com/q/295258/442
    $endgroup$
    – GEdgar
    Apr 12 '18 at 12:50










  • $begingroup$
    In Graham, Knuth, and Patashnik's Concrete Mathematics they have a comment when introducing the hypergeometric series on the etymology of the name in relation to the geometric series. I do not have the book in front of me right now. Maybe someone else can check.
    $endgroup$
    – KCd
    Apr 12 '18 at 13:21








  • 2




    $begingroup$
    On Page 206 (Second Edition) they write: "... is called hypergeometric series because it includes the geometric series as a very special case."
    $endgroup$
    – p4sch
    Apr 12 '18 at 16:14
















4












$begingroup$


I was wondering where the term "hypergeometric" for the hypergeometric function $_2F_1(a, b; c; z)$ comes from. Wikipedia says that the term was coined by J. Wallis, but I couldn't find any (mathematical) reason why these functions are anything like hyper-geometric.



Does anyone know where this comes from?






terminology special-functions







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    mathworld.wolfram.com/HypergeometricFunction.html
    $endgroup$
    – Karn Watcharasupat
    Apr 12 '18 at 8:42






  • 1




    $begingroup$
    see also math.stackexchange.com/q/295258/442
    $endgroup$
    – GEdgar
    Apr 12 '18 at 12:50










  • $begingroup$
    In Graham, Knuth, and Patashnik's Concrete Mathematics they have a comment when introducing the hypergeometric series on the etymology of the name in relation to the geometric series. I do not have the book in front of me right now. Maybe someone else can check.
    $endgroup$
    – KCd
    Apr 12 '18 at 13:21








  • 2




    $begingroup$
    On Page 206 (Second Edition) they write: "... is called hypergeometric series because it includes the geometric series as a very special case."
    $endgroup$
    – p4sch
    Apr 12 '18 at 16:14














4












4








4


1



$begingroup$


I was wondering where the term "hypergeometric" for the hypergeometric function $_2F_1(a, b; c; z)$ comes from. Wikipedia says that the term was coined by J. Wallis, but I couldn't find any (mathematical) reason why these functions are anything like hyper-geometric.



Does anyone know where this comes from?






terminology special-functions







share|cite|improve this question











$endgroup$




I was wondering where the term "hypergeometric" for the hypergeometric function $_2F_1(a, b; c; z)$ comes from. Wikipedia says that the term was coined by J. Wallis, but I couldn't find any (mathematical) reason why these functions are anything like hyper-geometric.



Does anyone know where this comes from?






terminology special-functions




terminology special-functions






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Apr 12 '18 at 12:49









Jack D'Aurizio

291k33284666




291k33284666










asked Apr 12 '18 at 8:24









StevenSteven

2,0081733




2,0081733








  • 1




    $begingroup$
    mathworld.wolfram.com/HypergeometricFunction.html
    $endgroup$
    – Karn Watcharasupat
    Apr 12 '18 at 8:42






  • 1




    $begingroup$
    see also math.stackexchange.com/q/295258/442
    $endgroup$
    – GEdgar
    Apr 12 '18 at 12:50










  • $begingroup$
    In Graham, Knuth, and Patashnik's Concrete Mathematics they have a comment when introducing the hypergeometric series on the etymology of the name in relation to the geometric series. I do not have the book in front of me right now. Maybe someone else can check.
    $endgroup$
    – KCd
    Apr 12 '18 at 13:21








  • 2




    $begingroup$
    On Page 206 (Second Edition) they write: "... is called hypergeometric series because it includes the geometric series as a very special case."
    $endgroup$
    – p4sch
    Apr 12 '18 at 16:14














  • 1




    $begingroup$
    mathworld.wolfram.com/HypergeometricFunction.html
    $endgroup$
    – Karn Watcharasupat
    Apr 12 '18 at 8:42






  • 1




    $begingroup$
    see also math.stackexchange.com/q/295258/442
    $endgroup$
    – GEdgar
    Apr 12 '18 at 12:50










  • $begingroup$
    In Graham, Knuth, and Patashnik's Concrete Mathematics they have a comment when introducing the hypergeometric series on the etymology of the name in relation to the geometric series. I do not have the book in front of me right now. Maybe someone else can check.
    $endgroup$
    – KCd
    Apr 12 '18 at 13:21








  • 2




    $begingroup$
    On Page 206 (Second Edition) they write: "... is called hypergeometric series because it includes the geometric series as a very special case."
    $endgroup$
    – p4sch
    Apr 12 '18 at 16:14








1




1




$begingroup$
mathworld.wolfram.com/HypergeometricFunction.html
$endgroup$
– Karn Watcharasupat
Apr 12 '18 at 8:42




$begingroup$
mathworld.wolfram.com/HypergeometricFunction.html
$endgroup$
– Karn Watcharasupat
Apr 12 '18 at 8:42




1




1




$begingroup$
see also math.stackexchange.com/q/295258/442
$endgroup$
– GEdgar
Apr 12 '18 at 12:50




$begingroup$
see also math.stackexchange.com/q/295258/442
$endgroup$
– GEdgar
Apr 12 '18 at 12:50












$begingroup$
In Graham, Knuth, and Patashnik's Concrete Mathematics they have a comment when introducing the hypergeometric series on the etymology of the name in relation to the geometric series. I do not have the book in front of me right now. Maybe someone else can check.
$endgroup$
– KCd
Apr 12 '18 at 13:21






$begingroup$
In Graham, Knuth, and Patashnik's Concrete Mathematics they have a comment when introducing the hypergeometric series on the etymology of the name in relation to the geometric series. I do not have the book in front of me right now. Maybe someone else can check.
$endgroup$
– KCd
Apr 12 '18 at 13:21






2




2




$begingroup$
On Page 206 (Second Edition) they write: "... is called hypergeometric series because it includes the geometric series as a very special case."
$endgroup$
– p4sch
Apr 12 '18 at 16:14




$begingroup$
On Page 206 (Second Edition) they write: "... is called hypergeometric series because it includes the geometric series as a very special case."
$endgroup$
– p4sch
Apr 12 '18 at 16:14










2 Answers
2






active

oldest

votes


















1












$begingroup$

The probability mass function of the hypergeometric distribution (related to the binomial distribution and the geometric distribution) is given by a ratio of products of binomial coefficients, just like the coefficients of the MacLaurin series of a hypergeometric function. This is my bet on the reason for picking the adjective hypergeometric for describing the $phantom{}_p F_q$ functions.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    It would be nice to know if the downvote is due to the fact that I am wrong, or something else.
    $endgroup$
    – Jack D'Aurizio
    Apr 12 '18 at 12:48










  • $begingroup$
    So the etymology is "geometric sequence" $to$ "geometric distribution" $to$ "hypergeometric distribution" $to$ "hypergeometric function"? I found it slightly disorienting that the answer started in the middle of that chain rather than at one of the ends -- but not so disorienting that I'd downvote.
    $endgroup$
    – Henning Makholm
    Apr 12 '18 at 12:59












  • $begingroup$
    @HenningMakholm: I am not sure that is the historical chain of events, but is sounds likely, doesn't it?
    $endgroup$
    – Jack D'Aurizio
    Apr 12 '18 at 13:16



















1












$begingroup$

This is the reason I believe: a geometric sequence is defined as a sequence where the ratio of two consecutive numbers is a constant
$$
frac{a_{n+1}}{a_n} = x,.
$$
You can generalize this notion by assuming the ratio to be any rational function of $n$ instead
$$
frac{a_{n+1}}{a_n} = frac{P(n)}{Q(n)},.
$$
Any such rational function can be factorized and rewritten as
$$
frac{P(n)}{Q(n)} = frac{(n+a_1)cdots(n+a_p)}{(n+b_1)cdots(n+b_q)(n+1)}x,.
$$
If $Q(n)$ doesn't have an $n+1$ factor we can always say, for example, $a_p=1$, so we don't lose generality. With this parametrization it's easy to check that
$$
sum_{n=0}^infty a_n = {}_pF_q(a_1,ldots, a_p;b_1,ldots, b_q;x),.
$$
Hypergeometric then suggests that this is a generalization of the geometric sequence. More precisely
$$
frac{1}{1-x} = {}_1F_0(1;;x),.
$$



Source: http://mathworld.wolfram.com/HypergeometricFunction.html






share|cite|improve this answer









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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    1












    $begingroup$

    The probability mass function of the hypergeometric distribution (related to the binomial distribution and the geometric distribution) is given by a ratio of products of binomial coefficients, just like the coefficients of the MacLaurin series of a hypergeometric function. This is my bet on the reason for picking the adjective hypergeometric for describing the $phantom{}_p F_q$ functions.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      It would be nice to know if the downvote is due to the fact that I am wrong, or something else.
      $endgroup$
      – Jack D'Aurizio
      Apr 12 '18 at 12:48










    • $begingroup$
      So the etymology is "geometric sequence" $to$ "geometric distribution" $to$ "hypergeometric distribution" $to$ "hypergeometric function"? I found it slightly disorienting that the answer started in the middle of that chain rather than at one of the ends -- but not so disorienting that I'd downvote.
      $endgroup$
      – Henning Makholm
      Apr 12 '18 at 12:59












    • $begingroup$
      @HenningMakholm: I am not sure that is the historical chain of events, but is sounds likely, doesn't it?
      $endgroup$
      – Jack D'Aurizio
      Apr 12 '18 at 13:16
















    1












    $begingroup$

    The probability mass function of the hypergeometric distribution (related to the binomial distribution and the geometric distribution) is given by a ratio of products of binomial coefficients, just like the coefficients of the MacLaurin series of a hypergeometric function. This is my bet on the reason for picking the adjective hypergeometric for describing the $phantom{}_p F_q$ functions.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      It would be nice to know if the downvote is due to the fact that I am wrong, or something else.
      $endgroup$
      – Jack D'Aurizio
      Apr 12 '18 at 12:48










    • $begingroup$
      So the etymology is "geometric sequence" $to$ "geometric distribution" $to$ "hypergeometric distribution" $to$ "hypergeometric function"? I found it slightly disorienting that the answer started in the middle of that chain rather than at one of the ends -- but not so disorienting that I'd downvote.
      $endgroup$
      – Henning Makholm
      Apr 12 '18 at 12:59












    • $begingroup$
      @HenningMakholm: I am not sure that is the historical chain of events, but is sounds likely, doesn't it?
      $endgroup$
      – Jack D'Aurizio
      Apr 12 '18 at 13:16














    1












    1








    1





    $begingroup$

    The probability mass function of the hypergeometric distribution (related to the binomial distribution and the geometric distribution) is given by a ratio of products of binomial coefficients, just like the coefficients of the MacLaurin series of a hypergeometric function. This is my bet on the reason for picking the adjective hypergeometric for describing the $phantom{}_p F_q$ functions.






    share|cite|improve this answer









    $endgroup$



    The probability mass function of the hypergeometric distribution (related to the binomial distribution and the geometric distribution) is given by a ratio of products of binomial coefficients, just like the coefficients of the MacLaurin series of a hypergeometric function. This is my bet on the reason for picking the adjective hypergeometric for describing the $phantom{}_p F_q$ functions.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Apr 12 '18 at 10:25









    Jack D'AurizioJack D'Aurizio

    291k33284666




    291k33284666












    • $begingroup$
      It would be nice to know if the downvote is due to the fact that I am wrong, or something else.
      $endgroup$
      – Jack D'Aurizio
      Apr 12 '18 at 12:48










    • $begingroup$
      So the etymology is "geometric sequence" $to$ "geometric distribution" $to$ "hypergeometric distribution" $to$ "hypergeometric function"? I found it slightly disorienting that the answer started in the middle of that chain rather than at one of the ends -- but not so disorienting that I'd downvote.
      $endgroup$
      – Henning Makholm
      Apr 12 '18 at 12:59












    • $begingroup$
      @HenningMakholm: I am not sure that is the historical chain of events, but is sounds likely, doesn't it?
      $endgroup$
      – Jack D'Aurizio
      Apr 12 '18 at 13:16


















    • $begingroup$
      It would be nice to know if the downvote is due to the fact that I am wrong, or something else.
      $endgroup$
      – Jack D'Aurizio
      Apr 12 '18 at 12:48










    • $begingroup$
      So the etymology is "geometric sequence" $to$ "geometric distribution" $to$ "hypergeometric distribution" $to$ "hypergeometric function"? I found it slightly disorienting that the answer started in the middle of that chain rather than at one of the ends -- but not so disorienting that I'd downvote.
      $endgroup$
      – Henning Makholm
      Apr 12 '18 at 12:59












    • $begingroup$
      @HenningMakholm: I am not sure that is the historical chain of events, but is sounds likely, doesn't it?
      $endgroup$
      – Jack D'Aurizio
      Apr 12 '18 at 13:16
















    $begingroup$
    It would be nice to know if the downvote is due to the fact that I am wrong, or something else.
    $endgroup$
    – Jack D'Aurizio
    Apr 12 '18 at 12:48




    $begingroup$
    It would be nice to know if the downvote is due to the fact that I am wrong, or something else.
    $endgroup$
    – Jack D'Aurizio
    Apr 12 '18 at 12:48












    $begingroup$
    So the etymology is "geometric sequence" $to$ "geometric distribution" $to$ "hypergeometric distribution" $to$ "hypergeometric function"? I found it slightly disorienting that the answer started in the middle of that chain rather than at one of the ends -- but not so disorienting that I'd downvote.
    $endgroup$
    – Henning Makholm
    Apr 12 '18 at 12:59






    $begingroup$
    So the etymology is "geometric sequence" $to$ "geometric distribution" $to$ "hypergeometric distribution" $to$ "hypergeometric function"? I found it slightly disorienting that the answer started in the middle of that chain rather than at one of the ends -- but not so disorienting that I'd downvote.
    $endgroup$
    – Henning Makholm
    Apr 12 '18 at 12:59














    $begingroup$
    @HenningMakholm: I am not sure that is the historical chain of events, but is sounds likely, doesn't it?
    $endgroup$
    – Jack D'Aurizio
    Apr 12 '18 at 13:16




    $begingroup$
    @HenningMakholm: I am not sure that is the historical chain of events, but is sounds likely, doesn't it?
    $endgroup$
    – Jack D'Aurizio
    Apr 12 '18 at 13:16











    1












    $begingroup$

    This is the reason I believe: a geometric sequence is defined as a sequence where the ratio of two consecutive numbers is a constant
    $$
    frac{a_{n+1}}{a_n} = x,.
    $$
    You can generalize this notion by assuming the ratio to be any rational function of $n$ instead
    $$
    frac{a_{n+1}}{a_n} = frac{P(n)}{Q(n)},.
    $$
    Any such rational function can be factorized and rewritten as
    $$
    frac{P(n)}{Q(n)} = frac{(n+a_1)cdots(n+a_p)}{(n+b_1)cdots(n+b_q)(n+1)}x,.
    $$
    If $Q(n)$ doesn't have an $n+1$ factor we can always say, for example, $a_p=1$, so we don't lose generality. With this parametrization it's easy to check that
    $$
    sum_{n=0}^infty a_n = {}_pF_q(a_1,ldots, a_p;b_1,ldots, b_q;x),.
    $$
    Hypergeometric then suggests that this is a generalization of the geometric sequence. More precisely
    $$
    frac{1}{1-x} = {}_1F_0(1;;x),.
    $$



    Source: http://mathworld.wolfram.com/HypergeometricFunction.html






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      This is the reason I believe: a geometric sequence is defined as a sequence where the ratio of two consecutive numbers is a constant
      $$
      frac{a_{n+1}}{a_n} = x,.
      $$
      You can generalize this notion by assuming the ratio to be any rational function of $n$ instead
      $$
      frac{a_{n+1}}{a_n} = frac{P(n)}{Q(n)},.
      $$
      Any such rational function can be factorized and rewritten as
      $$
      frac{P(n)}{Q(n)} = frac{(n+a_1)cdots(n+a_p)}{(n+b_1)cdots(n+b_q)(n+1)}x,.
      $$
      If $Q(n)$ doesn't have an $n+1$ factor we can always say, for example, $a_p=1$, so we don't lose generality. With this parametrization it's easy to check that
      $$
      sum_{n=0}^infty a_n = {}_pF_q(a_1,ldots, a_p;b_1,ldots, b_q;x),.
      $$
      Hypergeometric then suggests that this is a generalization of the geometric sequence. More precisely
      $$
      frac{1}{1-x} = {}_1F_0(1;;x),.
      $$



      Source: http://mathworld.wolfram.com/HypergeometricFunction.html






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        This is the reason I believe: a geometric sequence is defined as a sequence where the ratio of two consecutive numbers is a constant
        $$
        frac{a_{n+1}}{a_n} = x,.
        $$
        You can generalize this notion by assuming the ratio to be any rational function of $n$ instead
        $$
        frac{a_{n+1}}{a_n} = frac{P(n)}{Q(n)},.
        $$
        Any such rational function can be factorized and rewritten as
        $$
        frac{P(n)}{Q(n)} = frac{(n+a_1)cdots(n+a_p)}{(n+b_1)cdots(n+b_q)(n+1)}x,.
        $$
        If $Q(n)$ doesn't have an $n+1$ factor we can always say, for example, $a_p=1$, so we don't lose generality. With this parametrization it's easy to check that
        $$
        sum_{n=0}^infty a_n = {}_pF_q(a_1,ldots, a_p;b_1,ldots, b_q;x),.
        $$
        Hypergeometric then suggests that this is a generalization of the geometric sequence. More precisely
        $$
        frac{1}{1-x} = {}_1F_0(1;;x),.
        $$



        Source: http://mathworld.wolfram.com/HypergeometricFunction.html






        share|cite|improve this answer









        $endgroup$



        This is the reason I believe: a geometric sequence is defined as a sequence where the ratio of two consecutive numbers is a constant
        $$
        frac{a_{n+1}}{a_n} = x,.
        $$
        You can generalize this notion by assuming the ratio to be any rational function of $n$ instead
        $$
        frac{a_{n+1}}{a_n} = frac{P(n)}{Q(n)},.
        $$
        Any such rational function can be factorized and rewritten as
        $$
        frac{P(n)}{Q(n)} = frac{(n+a_1)cdots(n+a_p)}{(n+b_1)cdots(n+b_q)(n+1)}x,.
        $$
        If $Q(n)$ doesn't have an $n+1$ factor we can always say, for example, $a_p=1$, so we don't lose generality. With this parametrization it's easy to check that
        $$
        sum_{n=0}^infty a_n = {}_pF_q(a_1,ldots, a_p;b_1,ldots, b_q;x),.
        $$
        Hypergeometric then suggests that this is a generalization of the geometric sequence. More precisely
        $$
        frac{1}{1-x} = {}_1F_0(1;;x),.
        $$



        Source: http://mathworld.wolfram.com/HypergeometricFunction.html







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 15 '18 at 22:31









        Mane.andreaMane.andrea

        1265




        1265






























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            In PowerPoint, is there a keyboard shortcut for bulleted / numbered list?

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            .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ height:90px;width:728px;box-sizing:border-box; } 2 OneNote has two useful keyboard shortcuts for bulleted and numbered lists, namely Ctrl + . and Ctrl + - . Is there something similar for PowerPoint where the work with lists is also very common? Haven't found anything here: Use keyboard shortcuts in PowerPoint. keyboard-shortcuts microsoft-powerpoint share | improve this question edited Aug 14 '14 at 14:49 Linger 2,826 10 28 40
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            How to put 3 figures in Latex with 2 figures side by side and 1 below these side by side images but in...

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            1 0 I want to put 3 images in LateX such as 2 figures are side by side horizotally and 3rd figure below these 2 side by side figures but in middle. I have following code: begin{figure}[H] centering begin{minipage}{.5textwidth} centering includegraphics[width=.50linewidth]{Text/Images/Genelec_8010_AP.jpg} captionof{figure}{Genelec 8010 AP} label{fig:Genelec 8010} end{minipage}% begin{minipage}{.5textwidth} centering includegraphics[width=.69linewidth]{Text/Images/Genelec_8020_CPM.jpg} captionof{figure}{Genelec 8020 CPM} label{fig:Genelec 8020} end{minipage}% end{figure} %%Here is my 3rd picture code begin{figure} centering includegraphics[width=5cm]{Text/Images/Genelec_8030_BPM.jpg} caption{Genelec 8030 BPM} label{fig:Genelec 8030} end{figure} I want to ask ho
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