# Tagged Questions

**7**

votes

**1**answer

487 views

### Has anyone seen this series?

I come across the following infinite series.
$$
\sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}.
$$
In particular, I am interested in the case where $a=1/4$.
...

**29**

votes

**4**answers

2k views

### Groups, quantum groups and (fill in the blank)

In the study of special functions there are three levels of objects, classical, basic and elliptic. These correspond to classical hypergeometric functions, basic (q-) hypergeometric functions, and ...

**27**

votes

**7**answers

2k views

### How should an analytic number theorist look at Bessel functions?

(And a related question: Where should an analytic number theorist learn about Bessel functions?)
Bessel functions occur quite frequently in analytic number theory. One example, Corollary 4.7 of ...

**18**

votes

**0**answers

1k views

### Trigonometry related to Rogers--Ramanujan identities

For integers $n\ge2$ and $k\ge2$, fix the notation
$$
[m]=\sin\frac{\pi m}{nk+1} \quad\text{and}\quad
[m]!=[1][2]\dots[m], \qquad m\in\mathbb Z_{>0}.
$$
Consider the following trigonometric ...

**14**

votes

**2**answers

1k views

### The function $\sum_{0}^{\infty} x^n/n^n$

The function $F(x) = \sum_{0}^{\infty} x^n/n^n$ may be familiar to many readers as an example sometimes used when teaching tests for absolute convergence of entire functions defined by power series. I ...

**2**

votes

**2**answers

385 views

### Resources for special functions, integral identities

In the past weeks, I have struggled with finding suitable tables for integral indentities for Beta functions, Chebyshev polynomials and their like.
I would like to ask for online/offline resources ...

**7**

votes

**2**answers

1k views

### Duality of eta product identities: a new idea?

Looking at the collection of Eta Function Product Identities by Michael Somos, it seems like generally those identities come in pairs:
let's call two eta product identities $\sum\limits_{i=1}^r ...

**3**

votes

**1**answer

457 views

### Pochhammer symbol of a differential, and hypergeometric polynomials

I have a minor result which I'm sure has come up somewhere before but I can't seem to find it.
Consider a confluent hypergeometric function of the form
$$\newcommand{\ff}{{}_1F_1}
...

**7**

votes

**4**answers

436 views

### Trig functions based on convex curves

Pardon my naivety, but I wonder if
much use has been found for
trigonometric functions
defined in terms of a centrally symmetric convex curve $K$ replacing
the circle $C$.
For example, here is the ...

**3**

votes

**0**answers

139 views

### Solvable parametric $7$th and $13$th degree equations using $\eta(q)/\eta(q^p)$?

Q: Why is that some polynomial relations between eta quotients have a solvable Galois group, even if the deg is $n>4$?
For example, we have the well-known modular equation,
$$u^6 - v^6 + ...