Taylor Series
Taylor series expansion around 0 gives
where γ is the Euler–Mascheroni constant. For k > 2, the coefficient ak for the zk term can be computed recursively as
where ζ(s) is the Riemann zeta function. For small values, this gives the following values:
| k | |
|---|---|
| 1 | 1.0000000000000000000000000000000000000000 |
| 2 | 0.5772156649015328606065120900824024310422 |
| 3 | −0.6558780715202538810770195151453904812798 |
| 4 | −0.0420026350340952355290039348754298187114 |
| 5 | 0.1665386113822914895017007951021052357178 |
| 6 | −0.0421977345555443367482083012891873913017 |
| 7 | −0.0096219715278769735621149216723481989754 |
| 8 | 0.0072189432466630995423950103404465727099 |
| 9 | −0.0011651675918590651121139710840183886668 |
| 10 | −0.0002152416741149509728157299630536478065 |
| 11 | 0.0001280502823881161861531986263281643234 |
| 12 | −0.0000201348547807882386556893914210218184 |
| 13 | −0.0000012504934821426706573453594738330922 |
| 14 | 0.0000011330272319816958823741296203307449 |
| 15 | −0.0000002056338416977607103450154130020573 |
| 16 | 0.0000000061160951044814158178624986828553 |
| 17 | 0.0000000050020076444692229300556650480600 |
| 18 | −0.0000000011812745704870201445881265654365 |
| 19 | 0.0000000001043426711691100510491540332312 |
| 20 | 0.0000000000077822634399050712540499373114 |
| 21 | −0.0000000000036968056186422057081878158781 |
| 22 | 0.0000000000005100370287454475979015481323 |
| 23 | −0.0000000000000205832605356650678322242954 |
| 24 | −0.0000000000000053481225394230179823700173 |
| 25 | 0.0000000000000012267786282382607901588938 |
| 26 | −0.0000000000000001181259301697458769513765 |
| 27 | 0.0000000000000000011866922547516003325798 |
| 28 | 0.0000000000000000014123806553180317815558 |
| 29 | −0.0000000000000000002298745684435370206592 |
| 30 | 0.0000000000000000000171440632192733743338 |
Read more about this topic: Reciprocal Gamma Function
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