Letting (so your ), I noticed

...

Now look at the Chebyshev polynomials for odd ...

...

So . Then,

.

This, I suppose, is as "close as we can get to something in the 'usual' toolbox", at least if your "usual" toolbox has enough to include well-known named sequences of polynomials like the Chebyshev polynomials.

Note also that

and, for ,

. Now

Taking and using the correspondence between hyperbolic and trigonometric functions gives . Also, for cosh we get . Thus the result above simplifies to (the drops out due to the evenness of cosh) and so we have a formal proof of the relation to the Chebyshev polynomials we just gave.

...

Now look at the Chebyshev polynomials for odd ...

...

So . Then,

.

This, I suppose, is as "close as we can get to something in the 'usual' toolbox", at least if your "usual" toolbox has enough to include well-known named sequences of polynomials like the Chebyshev polynomials.

Note also that

and, for ,

. Now

Taking and using the correspondence between hyperbolic and trigonometric functions gives . Also, for cosh we get . Thus the result above simplifies to (the drops out due to the evenness of cosh) and so we have a formal proof of the relation to the Chebyshev polynomials we just gave.