can′t find coefficient f^((n)) (α) of Y_ν (z) formal power series of Y_ν (z) is Y_ν (z)=Σ_(h=0) ^∞ ((Y_ν ^((h)) (α))/(h!))(z−α)^h But.. can′t generalize coeff Y_ν ^((h)) (α) series representation ↓ Y_ν (z)=−(1/π)((2/z))^ν ∙Σ_(h=0) ^(ν−1) ((𝚪(ν−h))/(h!))((z/2))^(2h) +(2/π)J_ν (z)ln((1/2)z)−(1/π)((z/2))^ν ∙Σ_(h=0) ^∞ (((−1)^h (ψ^((0)) (h+1)−ψ^((0)) (h+ν+1)))/(h!(h+ν)!))((z/2))^(2h)


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Question Number 213007 by issac last updated on 30/Oct/24

${can}^{'} t find coefficient f^{(n)} (α) of Y_{ν} (z)$ $formal power series of Y_{ν} (z) is$ $Y_{ν} (z) = \underset{h = 0}{\sum^{\infty}} \frac{Y_{ν}^{(h)} (α)}{h!} {(z - α)}^{h}$ $B u t . . {can}^{'} t generalize coeff Y_{ν}^{(h)} (α)$ $series representation ↓$ $Y_{ν} (z) = - \frac{1}{π} {(\frac{2}{z})}^{ν} \cdot \underset{h = 0}{\sum^{ν - 1}} \frac{Γ (ν - h)}{h!} {(\frac{z}{2})}^{2 h} + \frac{2}{π} J_{ν} (z) \ln (\frac{1}{2} z) - \frac{1}{π} {(\frac{z}{2})}^{ν} \cdot \underset{h = 0}{\sum^{\infty}} \frac{{(- 1)}^{h} (ψ^{(0)} (h + 1) - ψ^{(0)} (h + ν + 1))}{h! (h + ν)!} {(\frac{z}{2})}^{2 h}$
	Answered by MrGaster last updated on 03/Nov/24
	$Y_ν (z)=(1/π)((2/z))^ν ∙Σ_(h=0) ^(ν−1) ((Γ(ν−h))/(h!))((z/2))^(2h) +(2/π)J_ν (z)ln((1/z)z)−(1/π)((z/2))^ν ∙Σ_(h=0) ^∞ (((−1)^h (ψ^((0)) (h+1)−ψ^((0)) (h+ν+1))/(h!(h+ν)!))((z/2))^(2h) Y_ν (z)Σ_(h=0) ^∞ ((Y_ν ^((h)) (α))/(h!))(z−α)^h Y_ν ^((h)) (α)= { ((−(1/π)((2/α))^ν ((Γ(ν−h))/(h!))((α/2))^(2h) ,h=0.1,…,ν−1)),(((2/π)J_ν ^((h−ν)) (α)ln((1/2)α)−(1/π)((α/2))^ν (((−1)^(h−ν) (ψ^((0)) (h−ν+1)−ψ^((0)) (h+1)))/((h−ν)!(h+ν)!))((α/2))^(2(h−ν)) ,h=ν,ν+1,…)) :}$
	$Y_{ν} (z) = \frac{1}{π} {(\frac{2}{z})}^{ν} \cdot \underset{h = 0}{\sum^{ν - 1}} \frac{Γ (ν - h)}{h!} {(\frac{z}{2})}^{2 h} + \frac{2}{π} J_{ν} (z) \ln (\frac{1}{z} z) - \frac{1}{π} {(\frac{z}{2})}^{ν} \cdot \underset{h = 0}{\sum^{\infty}} \frac{{(- 1)}^{h} (ψ^{(0)} (h + 1) - ψ^{(0)} (h + ν + 1)}{h! (h + ν)!} {(\frac{z}{2})}^{2 h}$ $Y_{ν} (z) \underset{h = 0}{\sum^{\infty}} \frac{Y_{ν}^{(h)} (α)}{h!} {(z - α)}^{h}$ $Y_{ν}^{(h)} (α) = {\begin{cases} - \frac{1}{π} {(\frac{2}{α})}^{ν} \frac{Γ (ν - h)}{h!} {(\frac{α}{2})}^{2 h}, h = 0.1, \dots, ν - 1 \\ \frac{2}{π} J_{ν}^{(h - ν)} (α) \ln (\frac{1}{2} α) - \frac{1}{π} {(\frac{α}{2})}^{ν} \frac{{(- 1)}^{h - ν} (ψ^{(0)} (h - ν + 1) - ψ^{(0)} (h + 1))}{(h - ν)! (h + ν)!} {(\frac{α}{2})}^{2 (h - ν)}, h = ν, ν + 1, \dots \end{cases}$
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