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# Tag Archives: Special Functions

## On the dubious function

The *dubious function * is defined as follows : and

Evaluate the sum

## 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.

## Improper Gamma integral

Let denote the Euler’s Gamma function. Prove that

where .