Gamma and Related Functions

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Gamma and Related Functions

A brief proposal for providing a complete suite of gamma and related functions as PyTorch operators. Enjoy!

One of a five-part series of special functions issues:

• Gamma and Related Functions (Gamma and Related Functions #78065)
• Bessel and Related Functions (Bessel and Related Functions #76324)
• Orthogonal Polynomials (Orthogonal Polynomials #80152)
• Elliptic Functions and Integrals (Elliptic Functions and Integrals #80157)

Operators

• barnes_g
• beta
• binomial_coefficient
• double_factorial
• factorial
• falling_factorial
• gamma
• log_barnes_g
• log_beta
• log_binomial
• log_double_factorial
• log_factorial
• log_falling_factorial
• log_gamma
• log_gamma_sign
• log_quadruple_factorial
• log_rising_factorial
• log_triple_factorial
• lower_incomplete_gamma
• polygamma
• quadruple_factorial
• reciprocal_gamma
• rising_factorial
• triple_factotial
• upper_incomplete_gamma

Documentation

Beta function

beta(a, b, other, *, out=None) → Tensor

Beta function:

$${\displaystyle \mathrm{B}(a, b) = \int _{0}^{1}t^{a - 1}(1 - t)^{b - 1} dt}.$$

A key property of the beta function is its close relationship to the gamma function:

$${\displaystyle \mathrm{B}(a, b)={\frac{\Gamma(a)\Gamma(b)}{\Gamma(a + b)}}.}$$

The closeness of the relationship between the gamma and beta functions is frequently applied in calculus and statistics (e.g., beta and related probability distributions).

The beta function is also closely related to binomial coefficients. When $a$ or $b$ is a positive integer:

$${\displaystyle \mathrm{B}(a, b) = {\dfrac{(a - 1)! (b - 1)!}{(a + b - 1)!}} = \frac{\frac{a + b}{ab}}{\binom{a + b}{a}}.}$$

Binomial coefficient

binomial_coefficient(input, k, *, out=None) → Tensor

Binomial coefficient:

$${\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}.}$$

Digamma function

digamma(z, *, out=None) → Tensor

The logarithmic derivative of the gamma function (i.e., the first of the polygamma functions):

$${\displaystyle \psi(z) = {\frac {\mathrm {d} } {\mathrm {d} z} } \ln {\big (}\Gamma (z){\big )} = {\frac {\Gamma '(z)} {\Gamma (z)} }\sim \ln {z}- {\frac {1}{2z}}.}$$

The digamma function is related to the harmonic numbers by:

$${\displaystyle \psi (n) = H_{n - 1} - \gamma ,}$$

where $H_{0} = 0$ and gamma is the Euler–Mascheroni constant.

double_factorial(input, *, out=None) → Tensor

Double factorial function:

$$\text{input}!!.$$

factorial(input, *, out=None) → Tensor

Factorial function:

$$\text{input}!.$$

The product of all positive integers less than or equal to $n$.

Many notable functions and number sequences are related to the factorials, e.g., binomial coefficients; double, triple, and quadruple factorials; and falling factorials and rising factorials.

falling_factorial(input, *, out=None) → Tensor

Falling factorial function:

$$\begin{aligned} (x)_{n}=x^{\underline {n}}&=\overbrace {x(x-1)(x-2)\cdots (x-n+1)} ^{n{\text{ terms}}}\\ &=\prod _{k=1}^{n}(x-k+1)=\prod _{k=0}^{n-1}(x-k),. \end{aligned}$$

The falling factorial is extended to real values of n using the gamma function provided x and x + n are real numbers that are not negative integers.

gamma(input, *, out=None) → Tensor

Gamma function:

$$\Gamma(\text{input}).$$

The standard extension of the factorial function to all complex numbers except non-positive integers.

log_beta(input, other, *, out=None) → Tensor

Natural logarithm of the Euler beta function:

$$\ln\text{B}(\text{input}, \text{other}).$$

log_double_factorial(input, *, out=None) → Tensor

Natural logarithm of the double factorial function:

$$\text{input}!!.$$

log_factorial(input, *, out=None) → Tensor

log_falling_factorial(input, *, out=None) → Tensor

log_gamma(input, *, out=None) → Tensor

Natural logarithm of the gamma function:

$$\log\Gamma(\text{input}).$$

log_quadruple_factorial(input, *, out=None) → Tensor

Natural logarithm of the quadruple factorial function:

$$\text{input}!!.$$

log_rising_factorial(input, *, out=None) → Tensor

log_triple_factorial(input, *, out=None) → Tensor

Natural logarithm of the triple factorial function:

$$\text{input}!!.$$

lower_incomplete_gamma(input, a, *, out=None) → Tensor

Lower incomplete gamma function:

$$\Gamma(a, \text{input}).$$

Incomplete gamma functions are special functions that arise as solutions to certain integrals. They are defined similarly to the gamma function but with incomplete integral limits. Unlike the gamma function, defined as an integral from zero to infinity, the lower incomplete gamma function is defined as an integral from $0$ to $a$.

upper_incomplete_gamma(input, a, *, out=None) → Tensor

Upper incomplete gamma function:

$$\Gamma(a, \text{input}).$$

Incomplete gamma functions are special functions that arise as solutions to certain integrals. They are defined similarly to the gamma function but with incomplete integral limits. Unlike the gamma function, defined as an integral from zero to infinity, the upper incomplete gamma function is defined as an integral from $a$ to infinity.

polygamma(input, n, *, out=None) → Tensor

Polygamma function:

$$\psi^{(n)}(\text{input}).$$

regularized_gamma(input, *, out=None) → Tensor

Regularized incomplete gamma function $Q(a, z)$.

inverse_regularized_gamma(input, *, out=None) → Tensor

Inverse of the regularized incomplete gamma function.

incomplete_beta(input, other, z, *, out=None) → Tensor

Incomplete Euler beta function $\text{B}_{z}(\text{input}, \text{other})$.

regularized_beta(input, *, out=None) → Tensor

Regularized incomplete beta function $I_{z}(a, b)$.

inverse_regularized_beta(input, *, out=None) → Tensor

Inverse of the regularized incomplete beta function.

triple_factorial(input, *, out=None) → Tensor

Triple factorial, $\text{input}!!!$.

quadruple_factorial(input, *, out=None) → Tensor

Quadruple factorial, $\text{input}!!!!$.

$${\displaystyle (x)_{n}={\frac {\Gamma (x+1)}{\Gamma (x-n+1)}},.}$$

log_falling_factorial(input, *, out=None) → Tensor

rising_factorial(input, *, out=None) → Tensor

$$ \begin{aligned}x^{(n)}=x^{\overline {n}}&=\overbrace {x(x+1)(x+2)\cdots (x+n-1)} ^{n{\text{ terms}}}\\ &=\prod _{k=1}^{n}(x+k-1)=\prod _{k=0}^{n-1}(x+k),. \end{aligned}$$

The rising factorial is extended to real values of n using the gamma function provided x and x + n are real numbers that are not negative integers:

$${\displaystyle x^{(n)}={\frac {\Gamma (x+n)}{\Gamma (x)}},.}$$

log_rising_factorial(input, *, out=None) → Tensor

barnes_g(input, *, out=None) → Tensor

Barnes G-function $G(z)$.

ln_barnes_g(input, *, out=None) → Tensor

Natural logarithm of the Barnes G-function $\ln G(z)$.

Notes

• multiple factorial operators are proposed (i.e., double, triple, and quadruple) to simplify differentiation;
• I’m unclear whether it’s possible to implement reentrant functions using the Jiterator;
• incomplete_gamma is implemented as gammainc and gammaincc. I recommend renaming gammainc to incomplete_gamma for clarity and consistency. I also recommend removing gammaincc in favor of regularized_gamma.

cc @mruberry @kshitij12345

[g-kakareko]

You can find the implementation here:
#41637 (comment)
The derivative calculation is based on the series expansion described in: http://www.jstor.org/stable/2348014

[g-kakareko]

What's the status of this issue? I hope the implementation I provided in the past was helpful?

[baxtrax]

I am also interested in the status, the gamma(s) function would be especially useful.