Laguerre–Pólya class

The Laguerre–Pólya class is the class of entire functions consisting of those functions which are locally the limit of a series of polynomials whose roots are all real. ^[1] Any function of Laguerre–Pólya class is also of Pólya class.

The product of two functions in the class is also in the class, so the class constitutes a monoid under the operation of function multiplication.

Some properties of a function ${\displaystyle E(z)}$ in the Laguerre–Pólya class are:

• All roots are real.
• ${\displaystyle |E(x+iy)|=|E(x-iy)|}$ for x and y real.
• ${\displaystyle |E(x+iy)|}$ is a non-decreasing function of y for positive y.

A function is of Laguerre–Pólya class if and only if three conditions are met:

• The roots are all real.
• The nonzero zeros z_n satisfy

${\displaystyle \sum _{n}{\frac {1}{|z_{n}|^{2}}}}$ converges, with zeros counted according to their multiplicity)

• The function can be expressed in the form of a Hadamard product

${\displaystyle z^{m}e^{a+bz+cz^{2}}\prod _{n}\left(1-z/z_{n}\right)\exp(z/z_{n})}$

with b and c real and c non-positive. (The non-negative integer m will be positive if E(0)=0. Note that if the number of zeros is infinite one may have to define how to take the infinite product.)

Examples

Some examples are ${\displaystyle \sin(z),\cos(z),\exp(z),\exp(-z),{\text{and }}\exp(-z^{2}).}$

On the other hand, ${\displaystyle \sinh(z),\cosh(z),{\text{and }}\exp(z^{2})}$ are not in the Laguerre–Pólya class.

For example,

${\displaystyle \exp(-z^{2})=\lim _{n\to \infty }(1-z^{2}/n)^{n}.}$

Cosine can be done in more than one way. Here is one series of polynomials having all real roots:

${\displaystyle \cos z=\lim _{n\to \infty }((1+iz/n)^{n}+(1-iz/n)^{n})/2}$

And here is another:

${\displaystyle \cos z=\lim _{n\to \infty }\prod _{m=1}^{n}\left(1-{\frac {z^{2}}{((m-{\frac {1}{2}})\pi )^{2}}}\right)}$

This shows the buildup of the Hadamard product for cosine.

If we replace z² with z, we have another function in the class:

${\displaystyle \cos {\sqrt {z}}=\lim _{n\to \infty }\prod _{m=1}^{n}\left(1-{\frac {z}{((m-{\frac {1}{2}})\pi )^{2}}}\right)}$

Another example is the reciprocal gamma function 1/Γ(z). It is the limit of polynomials as follows:

${\displaystyle 1/\Gamma (z)=\lim _{n\to \infty }{\frac {1}{n!}}(1-(\ln n)z/n)^{n}\prod _{m=0}^{n}(z+m).}$

References