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3

expr = Hypergeometric1F1[-a, 1/2, x]; Use SeriesCoefficient to find the coefficient in the series expansion c[n_] = SeriesCoefficient[expr, {x, 0, n}] Verifying that the infinite sum is the original expression Sum[c[n]*x^n, {n, 0, Infinity}] == expr (* True *) As shown in the documentation for Hypergeometric1F1, the series expansion is Sum[Pochhammer[...


7

Recall that $(-n)_k=0$ for $k>n, n,k\in\mathbb N$. Thus, what you have is an appropriate truncation of the usual series for the Kummer function. With[{a = 9}, Sum[Pochhammer[1/2 - a/2, k] (-x)^k/(Pochhammer[3/2, k] k!), {k, 0, a/2 - 1/2}]] 1 + (8 x)/3 + (8 x^2)/5 + (32 x^3)/105 + (16 x^4)/945


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