几种级数函数及函数方程

一、f0(z)=∑(k=1…∞)(-1)^k-1ln(1+z/k)

             =f0(1)-f0(z+1)+ln(z+1).

             [f0(1)=ln(π/2)]

 

二、f1(z)=∑(k=1…∞)(-1)^k-1[k·ln(1+z/k)-z]

             =f1(1)-f1(z+1)+f0(z)-z.

 

三、f2(z)=∑(k=1…∞)(-1)^k-1[k²·ln(1+z/k)-kz+(1/2)z²]

             =f2(1)-f2(z+1)-f0(z)+2f1(z)+z+(1/2)z².

 

四、f3(z)=∑(k=1…∞)(-1)^k-1[k³·ln(1+z/k)-k²z+(k/2)z²-(1/3)z³].

             =f3(1)-f3(z+1)+f0(z)-3f1(z)+3f2(z)-z-(1/2)z²-(1/3)z³.

 

五、f4(z)=∑(k=1…∞)(-1)^k-1[k⁴·ln(1+z/k)-k³z+(k²/2)z²-(k/3)z³+(1/4)z⁴].

             =f4(1)-f4(z+1)-f0(z)+4f1(z)-6f2(z)+4f3(z)+z+(3/2)z²+z³+(1/4)z⁴.

 

六、f5(z)=（略）