Heinrich Eduard Heine worked on Legendre polynomials, Lamé functions and Bessel functions.

Heine-Sommerfeld functions are defined in terms of Bessel functions by the equations

He is best remembered for the Heine-Borel theorem which stated that a subset of the reals is compact if and only if it is closed and bounded. Heine also formulated the concept of uniform continuity.
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Felice Casorati, an Italian mathematician, recognized many of the analogous properties of difference and differential equations. The Casoratian is the difference equation analog to the Wronksian in differential equations.  He developed the Casorati formula for linear difference equations.

Using N⁺ or N_n0⁺ as definition set , denoting a sequence by { y_n }, which is the set of all values of the function y on  N_n0⁺ .

Definition 1
Let  p₀ (n) = 1, p₁ (n), ..., p_k (n), g_n be k + 2 functions defined on N_n0⁺ .  An equation of the form

With (1), the following k initial conditions

are associated where c_i are real or complex constants.

Definition 2
If for every n e N_n0⁺ , g(n) = 0, then the equation (1) is said to be homogenous.

Definition 3
The function  f_i (n), i= 1,2,...,k are linearly independent if for all n,
 

implies a_i = 0, i = 1,2,...,k.

A sufficient condition for the functions  f_i (n), i= 1,2,...,k, to be linearly independent is that there exists an N  n₀ such that det K(N)  0.

Let us now take k functions f_i (n) defined on N_n0⁺ and define the matrix
 

Any other solution y(n, n₀ , c) can be expressed as a linear combination of

The matrix K(n) is called the matrix of Casorati  (Casoratian) and it plays in the theory of difference equations the same role as the Wronskian (Wronski) matrix in the theory of linear differential equations.

Georg Friedrich Bernhard Riemann is known for using Riemann's zeta function to investigate the distribution of the prime numbers.

The Riemann zeta function z(s) is defined by
 

z(2n) is a rational multiple of  p²ⁿ :
 

where B_n is Bernoulli's number.  Thus
 

One of the most striking property of the zeta function, discovered by Riemann himself, is the functional equation :
 

From the continuation of z(s) in the half plane Â(s) > 0, notice that the functional equation is
gives the analytic continuation of z(s) to the whole complex plane.

This analytic continuation can be obtained in several ways.

The complement formula of the Gamma function entails the formula:

which gives a symetry of the functional equation with respect to the line Â(s) = 1/2.

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