Engineering Mathematics: Jul 24, 2013

Definition of Bell series

$B(x)= \sum _{n=0}^{\infty }B_{n}\frac{x^{n}}{n!}= =e^{e^{x}-1}$

BS.ID.1: Exponential representation

Given

$\sum _{n=0}^{\infty }\begin{Bmatrix}n\\ k \end{Bmatrix}\frac{x^{n}}{n!} = \sum _{n=0}^{\infty }\frac{1}{k!}\sum _{j=0}^{k}(-1)^{k-j}\binom{k}{j}j^{n}\frac{x^{n}}{n!}$

Proof

$\sum _{n=0}^{\infty }\begin{Bmatrix}n\\ k \end{Bmatrix}\frac{x^{n}}{n!} = \sum _{n=0}^{\infty }\frac{1}{k!n!}\left [\sum _{j=0}^{k}(-1)^{k-j}\binom{k}{j}j^{n}x^{n} \right ]$

(1)

$\sum _{n=0}^{\infty }\begin{Bmatrix}n\\ k \end{Bmatrix}\frac{x^{n}}{n!} = \sum _{n=0}^{\infty }\frac{(jx)^{n}}{n!}\left [\sum _{j=0}^{k}(-1)^{k-j}\binom{k}{j} \right ]\frac{1}{k!}$

(2)

$\sum _{n=0}^{\infty }\begin{Bmatrix}n\\ k \end{Bmatrix}\frac{x^{n}}{n!} = e^{jx}\left [\sum _{j=0}^{k}(-1)^{k-j}\binom{k}{j} \right ]\frac{1}{k!}$

(3)

$\sum _{n=0}^{\infty }\begin{Bmatrix}n\\ k \end{Bmatrix}\frac{x^{n}}{n!} = \left [\sum _{j=0}^{k}(-1)^{k-j}\binom{k}{j}e^{jx} \right ]\frac{1}{k!}$

(4)

Using binomial expansion

$(x-y)^{n} = \sum _{k=0}^{n}x^{k}(-y)^{n-k}=\sum _{k=0}^{n}x^{k}(-1)^{n-k}(y)^{n-k}$

(5)

Combining (4) and (5),

$\sum _{n=0}^{\infty }\begin{Bmatrix}n\\ k \end{Bmatrix}\frac{x^{n}}{n!} = \left [\sum _{j=0}^{k}(-1)^{k-j}\binom{k}{j}e^{jx} \right ]\frac{1}{k!} = \frac{(e^{x}-1)^{k}}{k!}$

(6)

$B_{n}=\sum _{k=0}^{n }\begin{Bmatrix}n\\ k \end{Bmatrix}$

(7)

$B(x)= \sum _{n=0}^{\infty }B_{n}\frac{x^{n}}{n!}= \sum _{n=0}^{\infty } \sum _{k=0}^{n }\begin{Bmatrix}n\\ k \end{Bmatrix} \frac{x^{n}}{n!}$

(8)

$B(x)= \sum _{n=0}^{\infty }B_{n}\frac{x^{n}}{n!}= \sum _{n=0}^{\infty } \sum _{k=0}^{n }\begin{Bmatrix}n\\ k \end{Bmatrix} \frac{x^{n}}{n!}= \sum _{k=0}^{n }\frac{(e^{x}-1)^{k}}{k!} =e^{e^{x}-1}$

(9)

$B(x)= \sum _{n=0}^{\infty }B_{n}\frac{x^{n}}{n!}= =e^{e^{x}-1}$

Wednesday, July 24, 2013

Bell series