# Series expansion with nth term of $\left(\frac{\sin (x)}{x}\right)^a$

Using Mathematica, I need an expansion with $$n$$th term of $$f(x)=\left(\frac{\sin (x)}{x}\right)^a$$ about $$x=0$$ where $$a\geq 0$$ or if $$f(x)=\sum_{n=0}^{\infty} b_{2n }x^{2n}$$ then I need a formula for $$b_{2n}$$ .

By wolfram alpha (see:https://www.wolframalpha.com/input?i=series+%28%28sin+x%29%2Fx%29%5Ea+at+x%3D0) we have $$f(x)=1-\frac{a}{6}x^2+\frac{a (5 a-2)}{360} x^4-\frac{a \left(35 a^2-42 a+16\right) }{45360}x^6+O(x^8)$$

Any help would be appreciated. Thank you.

• The question is??? You don't believe the result?
– Ulrich Neumann
Mar 22 at 12:01
• No, I believe the result. Question is the nth term of the expansion.
– Max
Mar 22 at 12:02
• Series[(Sin[x]/x)^a, {x, 0, 6}] // Normal // Map[Factor] ?
– Syed
Mar 22 at 12:09
• @Max Coefficient for variable n ?
– Ulrich Neumann
Mar 22 at 12:12
• Maple's best is $$\sum _{k=0}^{\infty } \left( {\frac {27\,{{\rm e}^{k\ln \left( 3 \right) }}\cos \left( 1/2\,k\pi \right) }{4\, \left( k+3 \right) !}}- 3/4\,{\frac {\cos \left( 1/2\,k\pi \right) }{ \left( k+3 \right) !}} \right) {x}^{k}$$ for $a=3$ by convert((sin(x)/x)^3, FPS). Maple fails with $a=4$. Mar 22 at 13:44

$$c_m(a)=\frac{1}{m}\sum_{k=1}^m(ka-m+k)\,\frac{(-1)^k}{(2k+1)!}\,c_{m-k}(a)$$ with $$c_0(a)=1$$.
May be, for a given value of $$a$$, something could be done but for general $$a$$ this could just be a dream.