Editing Digamma function (section)

===Series with Gregory's coefficients, Cauchy numbers and Bernoulli polynomials of the second kind===
There exist various series for the digamma containing rational coefficients only for the rational arguments. In particular, the series with [[Gregory coefficients|Gregory's coefficients]]  {{math|''G''<sub>''n''</sub>}} is
:<math>
\psi(v) =\ln v- \sum_{n=1}^\infty\frac{\big| G_{n}\big|(n-1)!}{(v)_{n}},\qquad 
\Re (v) >0,
</math>
: <math>
\psi(v) =2\ln\Gamma(v) - 2v\ln v + 2v +2\ln v -\ln2\pi - 2\sum_{n=1}^\infty\frac{\big|G_{n}(2)\big|}{(v)_{n}}\,(n-1)! ,\qquad 
\Re (v) >0,
</math>
: <math>
\psi(v) =3\ln\Gamma(v) - 6\zeta'(-1,v) + 3v^2\ln{v} -  \frac32 v^2 - 6v\ln(v)+  3 v+3\ln{v} - \frac32\ln2\pi + \frac12  
- 3\sum_{n=1}^\infty\frac{\big| G_{n}(3) \big|}{(v)_{n}}\,(n-1)! ,\qquad 
\Re (v) >0,
</math>
where {{math|(''v'')<sub>''n''</sub>}} is the ''[[Falling and rising factorials|rising factorial]]'' {{math|(''v'')<sub>''n''</sub> {{=}} 
''v''(''v''+1)(''v''+2) ... (''v''+''n''-1)}},  {{math|''G''<sub>''n''</sub>(''k'')}} are the [[Gregory coefficients]] of higher order with {{math|''G''<sub>''n''</sub>(1) {{=}} ''G''<sub>''n''</sub>}}, {{math|&Gamma;}} is the [[gamma function]] and {{math|&zeta;}} is the [[Hurwitz zeta function]].<ref name="blag2016">{{cite journal
 | last = Blagouchine | first = Ia. V.
 | arxiv = 1408.3902
 | journal = Journal of Mathematical Analysis and Applications
 | pages = 404–434
 | title = Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to {{math|&pi;<sup>−1</sup>}} 
 | volume = 442
 | year = 2016| bibcode = 2014arXiv1408.3902B
 | doi = 10.1016/J.JMAA.2016.04.032
 | s2cid = 119661147
 }}</ref><ref name="blag2018" />
Similar series with the Cauchy numbers of the second kind {{math|''C''<sub>''n''</sub>}} reads<ref name="blag2016" /><ref name="blag2018" />
:<math>
\psi(v)=\ln(v-1) + \sum_{n=1}^\infty\frac{C_{n}(n-1)!}{(v)_{n}},\qquad 
\Re(v) >1,
</math>
A series with the [[Bernoulli polynomials of the second kind]] has the following form<ref name="blag2018" />
:<math>
\psi(v)=\ln(v+a) + \sum_{n=1}^\infty\frac{(-1)^n\psi_{n}(a)\,(n-1)!}{(v)_{n}},\qquad \Re(v)>-a,
</math>
where {{math|''ψ<sub>n</sub>''(''a'')}} are the ''Bernoulli polynomials of the second kind'' defined by the generating
equation
: <math>
\frac{z(1+z)^a}{\ln(1+z)}= \sum_{n=0}^\infty z^n \psi_n(a) \,,\qquad |z|<1\,,
</math>
It may be generalized to
: <math>
\psi(v)= \frac{1}{r}\sum_{l=0}^{r-1}\ln(v+a+l) + \frac{1}{r}\sum_{n=1}^\infty\frac{(-1)^n N_{n,r}(a)(n-1)!}{(v)_{n}},
\qquad \Re(v)>-a, \quad r=1,2,3,\ldots
</math>
where the polynomials {{math|''N<sub>n,r</sub>''(''a'')}} are given by the following generating equation
: <math>
\frac{(1+z)^{a+m}-(1+z)^{a}}{\ln(1+z)}=\sum_{n=0}^\infty N_{n,m}(a) z^n , \qquad |z|<1,
</math>
so that {{math|''N<sub>n,1</sub>''(''a'') {{=}} ''ψ<sub>n</sub>''(''a'')}}.<ref name="blag2018" /> Similar expressions with the logarithm of the gamma function involve these formulas<ref name="blag2018" />
: <math>
\psi(v)= \frac{1}{v+a-\tfrac12}\left\{\ln\Gamma(v+a) + v - \frac12\ln2\pi - \frac12 + \sum_{n=1}^\infty\frac{(-1)^n \psi_{n+1}(a)}{(v)_{n}}(n-1)!\right\},\qquad \Re(v)>-a, 
</math>
and
: <math>
\psi(v)= \frac{1}{\tfrac{1}{2}r+v+a-1}\left\{\ln\Gamma(v+a) + v - \frac12\ln2\pi - \frac12 + \frac{1}{r}\sum_{n=0}^{r-2} (r-n-1)\ln(v+a+n) +\frac{1}{r}\sum_{n=1}^\infty\frac{(-1)^n N_{n+1,r}(a)}{(v)_{n}}(n-1)!\right\},
</math>
where <math>\Re(v)>-a</math> and <math>r=2,3,4,\ldots</math>.