Is there a closed formula for $sum_{k=0}^n binom{alpha}{n-k} binom{beta}{k}(-1)^k$?












2












$begingroup$


Suppose $alpha, beta, n$ are three non-negative integers with $nleq min(alpha, beta)$. Is there a closed formula (or a combinatorical concept) for
$$
C_n^{(alpha, beta)}:=sum_{k=0}^n binom{n}{k} (-1)^k (alpha)_{n-k}(beta)_{k}=n!
sum_{k=0}^n binom{alpha}{n-k} binom{beta}{k}(-1)^k
$$
here $(x)_k=x(x-1)cdots(x-k+1)$ is the falling factorial. If that $(-1)^k$ was not there then the formula would've been $(alpha+beta)_n$. Also, is there a combinatorical intepretation of this number?






combinatorics binomial-coefficients







share|cite|improve this question









$endgroup$

















    2












    $begingroup$


    Suppose $alpha, beta, n$ are three non-negative integers with $nleq min(alpha, beta)$. Is there a closed formula (or a combinatorical concept) for
    $$
    C_n^{(alpha, beta)}:=sum_{k=0}^n binom{n}{k} (-1)^k (alpha)_{n-k}(beta)_{k}=n!
    sum_{k=0}^n binom{alpha}{n-k} binom{beta}{k}(-1)^k
    $$
    here $(x)_k=x(x-1)cdots(x-k+1)$ is the falling factorial. If that $(-1)^k$ was not there then the formula would've been $(alpha+beta)_n$. Also, is there a combinatorical intepretation of this number?






    combinatorics binomial-coefficients







    share|cite|improve this question









    $endgroup$















      2












      2








      2


      0



      $begingroup$


      Suppose $alpha, beta, n$ are three non-negative integers with $nleq min(alpha, beta)$. Is there a closed formula (or a combinatorical concept) for
      $$
      C_n^{(alpha, beta)}:=sum_{k=0}^n binom{n}{k} (-1)^k (alpha)_{n-k}(beta)_{k}=n!
      sum_{k=0}^n binom{alpha}{n-k} binom{beta}{k}(-1)^k
      $$
      here $(x)_k=x(x-1)cdots(x-k+1)$ is the falling factorial. If that $(-1)^k$ was not there then the formula would've been $(alpha+beta)_n$. Also, is there a combinatorical intepretation of this number?






      combinatorics binomial-coefficients







      share|cite|improve this question









      $endgroup$




      Suppose $alpha, beta, n$ are three non-negative integers with $nleq min(alpha, beta)$. Is there a closed formula (or a combinatorical concept) for
      $$
      C_n^{(alpha, beta)}:=sum_{k=0}^n binom{n}{k} (-1)^k (alpha)_{n-k}(beta)_{k}=n!
      sum_{k=0}^n binom{alpha}{n-k} binom{beta}{k}(-1)^k
      $$
      here $(x)_k=x(x-1)cdots(x-k+1)$ is the falling factorial. If that $(-1)^k$ was not there then the formula would've been $(alpha+beta)_n$. Also, is there a combinatorical intepretation of this number?






      combinatorics binomial-coefficients




      combinatorics binomial-coefficients






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 18 '18 at 23:55









      HamedHamed

      4,813622




      4,813622






















          1 Answer
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          $begingroup$

          Maple gets
          $$ {alphachoose n}{mbox{$_2$F$_1$}(-n,-beta;,alpha-n+1;,-1)}$$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            What does the notation ₂F₁ mean?
            $endgroup$
            – Gregory Nisbet
            Dec 19 '18 at 0:59








          • 1




            $begingroup$
            @GregoryNisbet Hypergeometric function.
            $endgroup$
            – Kemono Chen
            Dec 19 '18 at 1:00











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          1 Answer
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          0












          $begingroup$

          Maple gets
          $$ {alphachoose n}{mbox{$_2$F$_1$}(-n,-beta;,alpha-n+1;,-1)}$$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            What does the notation ₂F₁ mean?
            $endgroup$
            – Gregory Nisbet
            Dec 19 '18 at 0:59








          • 1




            $begingroup$
            @GregoryNisbet Hypergeometric function.
            $endgroup$
            – Kemono Chen
            Dec 19 '18 at 1:00
















          0












          $begingroup$

          Maple gets
          $$ {alphachoose n}{mbox{$_2$F$_1$}(-n,-beta;,alpha-n+1;,-1)}$$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            What does the notation ₂F₁ mean?
            $endgroup$
            – Gregory Nisbet
            Dec 19 '18 at 0:59








          • 1




            $begingroup$
            @GregoryNisbet Hypergeometric function.
            $endgroup$
            – Kemono Chen
            Dec 19 '18 at 1:00














          0












          0








          0





          $begingroup$

          Maple gets
          $$ {alphachoose n}{mbox{$_2$F$_1$}(-n,-beta;,alpha-n+1;,-1)}$$






          share|cite|improve this answer









          $endgroup$



          Maple gets
          $$ {alphachoose n}{mbox{$_2$F$_1$}(-n,-beta;,alpha-n+1;,-1)}$$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 19 '18 at 0:02









          Robert IsraelRobert Israel

          323k23213467




          323k23213467












          • $begingroup$
            What does the notation ₂F₁ mean?
            $endgroup$
            – Gregory Nisbet
            Dec 19 '18 at 0:59








          • 1




            $begingroup$
            @GregoryNisbet Hypergeometric function.
            $endgroup$
            – Kemono Chen
            Dec 19 '18 at 1:00


















          • $begingroup$
            What does the notation ₂F₁ mean?
            $endgroup$
            – Gregory Nisbet
            Dec 19 '18 at 0:59








          • 1




            $begingroup$
            @GregoryNisbet Hypergeometric function.
            $endgroup$
            – Kemono Chen
            Dec 19 '18 at 1:00
















          $begingroup$
          What does the notation ₂F₁ mean?
          $endgroup$
          – Gregory Nisbet
          Dec 19 '18 at 0:59






          $begingroup$
          What does the notation ₂F₁ mean?
          $endgroup$
          – Gregory Nisbet
          Dec 19 '18 at 0:59






          1




          1




          $begingroup$
          @GregoryNisbet Hypergeometric function.
          $endgroup$
          – Kemono Chen
          Dec 19 '18 at 1:00




          $begingroup$
          @GregoryNisbet Hypergeometric function.
          $endgroup$
          – Kemono Chen
          Dec 19 '18 at 1:00


















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