Let , and define on for with on .

Show that

Remark. In the case of constant function, i.e., on (provided that ) for some ,

we can compute that on for all , which turns out to be a well known result.

That is,

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- March 14th 2011, 12:04 PMbkarpuzGeneralized Taylor series of an exponential
Let , and define on for with on .

Show that

**Remark**. In the case of constant function, i.e., on (provided that ) for some ,

we can compute that on for all , which turns out to be a well known result.

That is,

- March 16th 2011, 12:19 PMgirdav
Let and . We can show that is analytic on the whole real line (for example by Faa-di-Bruno's formula) and we have according to the chain rule.

We show by induction that . so the result is true for and if it's true for n we have

hence

. We can now compare the two Taylor series.