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binomial_coefficient_over_the_complex_numbers [2016/05/12 19:00] nikolaj |
binomial_coefficient_over_the_complex_numbers [2016/08/21 11:33] nikolaj |
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| + | $(1+z)^s$ | ||
| + | |||
| + | Sum[Gamma[1 + x]/(Gamma[1 + x - y] Gamma[1 + y])z^y, {y, 0, \[Infinity]}] | ||
| + | |||
| === Discussion === | === Discussion === | ||
| For natural numbers $n\ge{k}$ we get | For natural numbers $n\ge{k}$ we get | ||
| Line 19: | Line 23: | ||
| From this it's also clear that ${n\choose{k}}$ is a sum of 1's, i.e. an integer. | From this it's also clear that ${n\choose{k}}$ is a sum of 1's, i.e. an integer. | ||
| + | |||
| + | <code> | ||
| + | (*Random generalization of linear such recursive schemes*) | ||
| + | |||
| + | f[m_, 0] = a[m]; | ||
| + | f[0, m_] = b[m]; | ||
| + | f[m_, m_] = c[m]; | ||
| + | |||
| + | f[n_, k_] := A*f[n, k - 1] + B*f[n - 1, k] + C*f[n - 1, k - 1] | ||
| + | |||
| + | Table[{n, k, f[n, k] // sexy}, {n, 1, 4}, {k, 1, 4}] // TableForm | ||
| + | </code> | ||
| === Reference === | === Reference === | ||
