I have read an article about Bernoulli Polynomial. I found that Bernoulli Polynomial has explicit formula like this:
$$B_n(x)=\sum_{k=0}^{\infty}\binom nk B_kx^{n-k}$$
And the article said that the explicit formula is obtained from Bernoulli Polynomial's generating function:
$$\frac{te^{xt}}{e^t-1}=\sum_{n=0}^{\infty}B_n(x)\frac{t^n}{n!}$$
My question is:
How to get that explicit formula of Bernoulli Polynomial by using its generating function?
I just thought about that its explicit formula was containt binomial theorem and application of Pascal's triangle.
Thank you so much for your helps.
Use $\quad t/(e^t-1) = \sum_{n=0}^\infty B_nt^n/n!\quad$ and $\quad e^{xt}=\sum_{n=0}^\infty x^n t^n/n!\quad$ and multiply the two exponential generating function power series in $t$. The resulting generating function power series coefficients is the binomial convolution which gives the formula for $B_n(x)$ in your first equation. Read the section on convolution in Wikipedia article about generating function for this.
By the way, the binomial theorem is one consequence of $\quad e^{(x+y)t}=e^{xt}e^{yt},\quad$ by using the binomial convolution of the coefficients to get $\quad (x+y)^n=\sum_{k=0}^n \binom nk x^k y^{n-k}.$