Lower fixed binomial coefficient → negative binomial theorem

$\left(\genfrac{}{}{0ex}{}{A+i}{A}\right)=[x^i]{\sum }_n\left(\genfrac{}{}{0ex}{}{A+n}{A}\right)x^n=[x^i]\frac{1}{(1-x{)}^{A+1}}$

• negative binomial theorem

• $(1-x{)}^{-d}={\sum }_{n=0}^{\infty }(\genfrac{}{}{0ex}{}{n+d-1}{d-1})x^n$

• binomial coefficient of under-fixation → negative binomial theorem

facing the opposite direction

• Power series → binomial coefficient :
• $[x^B]\frac{1}{(1-x{)}^A}=\left(\genfrac{}{}{0ex}{}{A+B-1}{A-1}\right)$

relevance

• Infinite sums with respect to the first argument of the binomial coefficients (mother function) - Skol’s Wisdom Book
  • This is generating function.

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