Is there a fractional version of Pascal's triangle for binomial expansion similar to the standard triangle used for binomial expansion? If so, is it related to the Gamma function? \begin{array}{cccccc} & & & 1 & & \\ & & 1 & & \frac{1}{2} & \\ & 1 & & \frac{1}{2} & & \frac{-1}{8} \\ 1 & & \frac{1}{2} & & \frac{-1}{8} & & \frac{1}{16} \\ \end{array}
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2$\begingroup$ You should provide context for your problem. What makes you think there is a connection between these topics? $\endgroup$– Kepler's TriangleCommented Nov 7 at 0:29
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Traditionally, Pascal's triangle (PT) contains binomial coefficients, defined only for non-negative integers in terms of combinations. Fractional Pascal's triangle (FPT) generalizes this concept by allowing $n$ to be any real or fractional number, not just integers. To this end, it uses fractional binomial coefficients, where instead of the standard formula for $\binom{n}{k}$, it uses the gamma function, replacing the factorials: $$ \binom{n}{k}=\frac{\Gamma(k+1)}{\Gamma(r−k+1)Γ(r+1)}. $$ Essentially, FPT allows for interpolation between integer rows of Pascal's triangle and generalizes the expansion $(1+x)^n$ for any real $n$.
