During the the proof of the following formula
I faced with $\phi(x)$ for negative integers. That is in order to finish proof of the mentioned formula (which is not proved in the book), after a long messy calculations I need to prove $$\dfrac{(-1)^{m+N} \phi(m-N)}{\Gamma(m+1-N)} = (N-m-1)!, \ \ m=0,1, \, \dots, \ N-1 \ \ (*)$$ to finish the final step but I stuck because $\phi(x)$ is not defined in the book for negative integers. For positive integers definition is $$\psi(n)= \dfrac{\Gamma'(n+1)}{\Gamma(n+1)} = -\gamma + \phi(n) = -\gamma + 1+ \frac12 + \dots + \frac1n. $$ So how equation $(*)$ holds?
I tried this way: $$\dfrac{(-1)^{m+N} \phi(m-N)}{\Gamma(m+1-N)} = \dfrac{(-1)^{m+N} [\psi(m-N) + \gamma]}{\Gamma(m+1-N)} = \dfrac{(-1)^{m+N} \psi(m-N)}{\Gamma(m+1-N)} = \dfrac{(-1)^{m+N} \Gamma'(m+1-N)}{\Gamma^2(m+1-N)}$$ but what is the value of the last term for negative integers, i.e. for when $m=0, 1, \dots, N-1$?