Pure Python special functions

Bessel–Clifford function.ipynb

| Bessel–Clifford function Cn (argument complex but n restricted to integer instead of complex)

q-analog.ipynb

| q-Pochhammer symbol q_poch

| q-bracket q_bracket

| q-factorial q_factorial

| q-binomial coefficients q_binom

| binomial coefficient binom (using q_binom and without factorial function)

| unilateral basic hypergeometric series q_hyper (accepts all parameters as complex numbers)

| bilateral basic hypergeometric series q_hyper(type=2) (accepts all parameters as complex numbers)

| q-gamma function q_gamma (when q<1, accepts argument as complex number, accepts base q>1)

| first Jackson q-Bessel function q_besselj1 (accepts base q as complex number)

| second Jackson q-Bessel function q_besselj2 (accepts argument and base q as complex numbers)

| third Jackson q-Bessel function q_besselj3 (accepts argument and base q as complex numbers)

| first modified q-Bessel function of the first kind q_besseli1 (accepts argument and base q as complex numbers)

| second modified q-Bessel function of the first kind q_besseli2 (accepts argument and base q as complex numbers)

| third modified q-Bessel function of the first kind q_besseli3 (accepts argument and base q as complex numbers)

| (not implemented) first modified q-Bessel function of the second kind q_besselk1 (not Pure Python, requires sin)

| (not implemented) second modified q-Bessel function of the second kind q_besselk2 (not Pure Python, requires sin)

| Jackson q-difference operator for q-Bessel funtions q_besselj1p q_besselj2p q_besselj3p q_besseli1p q_besseli2p q_besseli3p

| symmetric q-difference operator for q-Bessel funtions q_besselj1p(type=2) q_besselj2p q_besselj3p q_besseli1p q_besseli2p q_besseli3p

| (not implemented) forward difference operator

| (not implemented) backward difference operator

| q-exponential functions e_q and E_q

| q-sine functions q_sin and q_Sin

| q-cosine functions q_cos and q_Cos

subfactorial.ipynb demo only (see treefactorial.py for special functions)

treefactorial.py

| factorial

| binomial coefficient, faster version using treefactorial binom

| subfactorial, fast algorithm subfactorial