The denominator of  .
The generalized hypergeometric function of two variables (Kampe de Feriet function):  . 
The regularized generalized hypergeometric function of two variables (regularized Kampe de Feriet function):  . 
The Lauricella function A of  variables:  . 
The Lauricella function B of  variables:  . 
The Lauricella function C of  variables:  . 
The Lauricella function D of  variables:  .
The Meijer G function:  . The infinite contour of integration  separates the poles of  at  ,  from the poles of  at  ,  . Such a contour always exists in the cases  . There are three possibilities for the contour  : (i)  runs from γ-ⅈ ∞ to γ+ⅈ ∞ (where  ) so that all poles of  ,  are to the left of  , and all poles of  ,  are to the right of ℒ. This contour can be a straight line  if  (then  ). (In this case, the integral converges if  ,  . If  , then  must be real and positive and the additional condition  should be added.) (ii)  is a left loop, starting and ending at -∞ and encircling all poles of  , , once in the positive direction, but none of the poles of  ,  . In this case, the integral converges if  and one of the following conditions is satisfied: ◼  or  and  ◼  and  and  and  . (iii)  is a right loop, starting and ending at +∞ and encircling all poles of  ,  , once in the negative direction, but none of the poles of  ,  . In this case, the integral converges if  and one of the following conditions is satisfied: ◼  or  and  ◼  and  and  and  . Identities containing MeijerG
The number of unrestricted partitions (independent of the order and with repetitions allowed) of the positive integer  into a sum of strictly positive integers that add up to  :  . Identities containing PartitionsP 
The  prime number (the smallest integer greater than  that cannot be divided by any integer greater than 1 and smaller than itself):  . Identities containing Prime 
A Boolean function that tests whether the angular momentum quantum numbers are physically realizable: 
The number of ordered partitions (independent of the order and no repetitions allowed) of the positive integer  into a sum of strictly positive integers which add up to  :  . Identities containing PartitionsQ
The Gauss type hypergeometric function  : 
The values of the Weierstrass zeta function at the half-periods  :  .
The half‐periods for Weierstrass elliptic functions corresponding to the invariants  :
The generalized Dirac comb function Ш(x):  . | 
for even Mathieu functions 
with characteristic exponent 
and parameter 
, such that there exists a solution of the corresponding Mathieu differential equation 
that is of the form 
, where 
is an even function of 
with period 
.
: 
and 
: 
.
root of the equation 
: 
.
.
.
. The value of 
for which the elliptic integral of the first kind 
has the value 
:
.
(where 
): 
.
for odd Mathieu functions 
with characteristic exponent 
and parameter 
, such that there exists a solution of the corresponding Mathieu differential equation 
that is of the form 
, where 
is an odd function of 
with period 
.
Bell number: 
.
in 
: 
.
Bernoulli number: 
.
in 
: 
.
in 
: 
.
Norlund polynomial 
.
.
.
.
.
root of the equation 
: 
.
.
.
: 
.
.
in 
: 
.
Gegenbauer function in 
: 
. For the nonnegative integer 
, the function 
is a polynomial in 
.
Gegenbauer function in 
for parameter 
: 
. For the nonnegative integer 
, the function 
is a polynomial in 
.
.
for which the Jacobi elliptic function cd, such that 
.
and parameter 
.
of the even Mathieu function with characteristic value 
and parameter 
: 
.
.
⩵
.
.
such that 
.
.
.
.
.
.
.
.
.
.
such that 
.
.
.
.
.
‐function: 
.
‐function: 
.
.
such that 
.
.
.
such that 
.
.
such that 
.
: 
.
function at the half‐periods 
: 
.
function at the half‐periods 
: 
.
.
.
Euler number: 
in 
: 
.
: 
.
. The values 
such that 
.
: 
.
and 
: 
: 
.
.
such that 
.
.
such that 
.
.
such that 
.
.
.
Fibonacci number: 
.
in 
: 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
, together with their exponents.
: 
.
, see 
.
and 
, see 
.
for Weierstrass elliptic functions corresponding to the half‐periods 
: 
.
.
.
.
harmonic number: 
.
: 
.
Hermite function in 
: 
. For nonnegative integer 
it is a polynomial in 
.
.
.
.
separates the poles of 
at 
, 
from the poles of 
at points 
, 
.
: 
.
.
.
such that 
.
.
such that 
.
: 
.
: 
.
.
root of the equation 
: 
.
.
.
: 
.
.
.
.
Lucas number: 
.
Laguerre function in 
: 
. For nonnegative integer 
it is a polynomial in 
.
generalized Laguerre polynomial in 
for parameter 
: 
. For nonnegative integer 
it is a polynomial in 
.
.
.
.
: 
. For 
it is a dilogarithm function in 
.
.
: 
.
.
.
): 
): 
.
such that 
.
.
such that 
.
.
such that 
.
Legendre function in 
: 
. For nonnegative integer 
it is a polynomial in 
.
.
.
Jacobi function in 
for parameters 
and 
: 
. For nonnegative integer 
it is a polynomial in 
.
and parameters 
, 
, 
.
of the angular spheroidal function of the first kind with variable 
and parameters 
, 
, 
: 
.
of the module 
: 
.
of the nome 
: 
.
Legendre function of the second kind: 
.
associated
.
such that 
.
.
such that 
.
and parameters 
, 
, 
.
of the angular spheroidal function of the second kind with variable 
and parameters 
, 
, 
: 
.
and
: 
.
as a sum of 
squares of different positive or negative integers.
(where 
has period 
).
: 
.
: 
.
.
.
.
.
and parameters 
, 
, 
.
of the radial spheroidal function of the first kind with variable 
and parameters 
, 
, 
: 
.
and parameters 
, 
, 
.
of the radial spheroidal function of the second kind with variable 
and parameters 
, 
, 
: 
.
.
such that 
.
.
such that 
.
and parameter 
.
of the odd Mathieu function with characteristic value 
and parameter 
: 
.
.
.
.
.
: 
.
.
.
.
.
.
.
such that 
.
and order 
appearing in the relations between radial and angular spheroidal functions.
and order 
appearing in expansions of radial spheroidal function of the first kind around 
.
.
Chebyshev function of the first kind: 
. For nonnegative integer 
it is a polynomial in 
.
.
.
.
.
.
Chebyshev function of the second kind: 
. For nonnegative integer 
it is a polynomial in 
.
: 
.
such that 
.
sheet. The 
value of 
such that 
.
.
.
root of the equation 
: 
.
.
and 
for parameters 
and 
: 
.
.
.
.
.
.
: 
Stieltjes constant: 
.
.
.
.
.
.
needed to place the list elements in canonical order.
.
.
.
.
.
.
.
of the first elliptic theta function: 
.
.
of the second elliptic theta function: 
.
.
of the third elliptic theta function: 
.
.
of the fourth elliptic theta function: 
.
.
.
.
.
with symmetric Riemann modular matrix 
with positive definite imaginary part and vector 
is defined through 
, where 
means transposed to 
matrix (or vector) and 
ranges over all possible vectors in the 
-dimensional integer lattice: 
with characteristic 
, symmetric Riemann modular matrix 
with positive definite imaginary part and vector 
is defined through 
, where 
means transposed to 
matrix (or vector) and 
ranges over all possible vectors in the 
-dimensional integer lattice: 
such that for any 
with 
the congruence 
holds.
.
and order 
of the corresponding Sturm‐Liouville wave differential equation 
).
: 
. 
: 
.
.
.
nontrivial zero of the Riemann's zeta function 
on the critical half‐line 
: 
.
powers of the divisors of 
: 
.
: 
.
.
.
.
.
: 
and relatively prime to 
(the Euler totient function): 
.
: 
.
derivative of the digamma function: 
.
for Weierstrass elliptic functions corresponding to the invariants 
: 
.
of the Weierstrass elliptic function P: 
.
. The value of 
such that 
: 
.
. The value of 
such that 
and 
: 
.