By polynomial formula ( ∑ i ∈ m x i ) n = ∑ j i ∈ Z + ∑ j i = n ( n j 0 , … , j m − 1 ) ∏ i ∈ m x i j i where n ∈ N But what about n ∉ N ?

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By polynomial formula  (      ∑          i      ∈      m            x    i        )    n    =      ∑                                                                      j                i                            ∈                                                Z                                +                                                                        ∑                              j                i                            =              n                                                (                            n                                                  j                          0                                ,          …          ,                      j                          m              −              1                                            )        ∏          i      ∈      m            x    i                  j        i            where   n  ∈      N  But what about   n  ∉      N  ?

Marisa Colon · Accepted Answer

When n is not an integer, multivariate Taylor series expansion is to be applied. Let&#039;s expand   (  a  +  b  +  c      )          α       around   b  =  0 and   c  =  0. We will need:            ∂              p        +        q                    ∂              b        p            ∂              c        q              (  a  +  b  +  c      )          α        =      ∏          k      =      0              p      +      q      −      1        (  α  −  k  )  ⋅  (  a  +  b  +  c      )          α      −      p      −      q        =                    (                    α              p        +        q                            )              (  p  +  q  )  !  ⋅  (  a  +  b  +  c      )          α      −      p      −      q      Thus:  (  a  +  b  +  c      )          α        =      ∑          p      =      0        ∞        ∑          q      =      0        ∞                      (                    α              p        +        q                            )              (  p  +  q  )  !  ⋅  (  a      )          α      −      p      −      q                  b      p              p      !                  c      q              q      !        =      ∑          p      =      0        ∞        ∑          q      =      0        ∞                      (                    α              p        ,        q        ,        α        −        p        −        q                            )                  a          α      −      p      −      q            b    p        c    q  As you can see, in the case of   α  ∈      N  , the series terminates, and the expansion reduces to the tri-nomial expansion.

By polynomial formula (sum_(i in m)x_i)^n=sum_(j_i in ZZ^+)(nj_0,…,j_(m−1))prod_(i in m)x^(j_i)_i

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