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# Tag Archives: Series

## Bessel series

Let denote the Bessel function of the first kind. Prove that:

Solution

The Jacobi – Anger expansion tells us that

Hence by Parseval’s Theorem it follows that:

## Pell – Lucas series

The Pell – Lucas numbers are defined as follows and for every it holds that

Prove that

## Gamma infinite product

Prove that

Solution

Converting the product to a sum and using duplication formula for the gamma function and telescoping,

Using Stirling formula

we get that

## Digamma and Trigamma functions

Let and denote the digamma and trigamma functions respectively. Prove that:

where denotes the Euler – Mascheroni constant.

Solution

We begin with the recently discovered identity:

Letting we get that

Now combining this result here we conclude the exercise.

## Trigamma series

Let denote the trigamma function. Prove that

Solution

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Find out more at his Encyclopedia Page.