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Question Number 67520 by mathmax by abdo last updated on 28/Aug/19

l e t f (x, z) = \frac{z e^{x z}}{e^{z} - 1} (x a n d z f r o m C)

1) p r o v e t h a t f (x, z) = \sum_{n = 0}^{\infty} B_{n} (x) \frac{z^{n}}{n!}

w i t h B_{n} (x) i s a u n i t a i r e p o l y n o m e w i t h d e g r e n

d e t e r m i n e B_{n} (x) i n t e r m s o f B_{n} (n u m b e r o f b e r n o u l l i)

2) p r o v e t h a t B_{n}^{^{'}} (x) = {n B}_{n - 1} (x)

B_{n} (x + 1) - B_{n} (x) = {n x}^{n - 1}

p r o v e t h a t f (x, z) = f (1 - x, - z) a n d B_{n} (1 - x) = {(- 1)}^{n} B_{n} (x)