Special polynomials

Consider the following polynomials that converge to entire functions :

f1(z) = 1 + z

f2(z) = 1 + z + z^2/4

f3(z) = 1 + z + z^2/4 + z^3/54

f4(z) = 1 + z + z^2/4 + z^3/54 + z^4/2380

f5(z) = 1 + z + z^2/4 + z^3/54 + z^4/2380 + z^5/339923

...

f(z) = 1 + z + z^2/4 + z^3/54 + z^4/2380 + z^5/339923 + ...

a_0 = 1 , a_1 = 1 , a_2 = 1/4 , a_3 = 1/54 , a_4 = 1/2380 , a_5 = 1/339923

All these polynomials have decreasing positive unit fractions (a_n) as coefficients.

The next coefficient a_n is the largest unit fraction such that the function fn(z) has only (negative) REAL zero's  :

fn(z) = 0  ==> z is a negative real !

f(z) = 0  ==> z is a negative real !

Clearly f(z) is an entire function.

**

t1(z) = 1 + z

t2(z) = 1 + z + z^2/4

t3(z) = 1 + z  + z^2/(2!)^2 + z^3/(4!)^2

t4(z) = 1 + z  + z^2/(2!)^2 + z^3/(4!)^2 + z^4/(7!)^2  

t5(z) = 1 + z  + z^2/(2!)^2 + z^3/(4!)^2 + z^4/(7!)^2 + z^5/(11!)^2 

t6(z) = 1 + z  + z^2/(2!)^2 + z^3/(4!)^2 + z^4/(7!)^2 + z^5/(11!)^2 + z^6/(16!)^2 

t7(z) =  1 + z  + z^2/(2!)^2 + z^3/(4!)^2 + z^4/(7!)^2 + z^5/(11!)^2 + z^6/(16!)^2 + z^7/(22!)^2

...

t(z) = 1 + z  + z^2/(2!)^2 + z^3/(4!)^2 + z^4/(7!)^2 + z^5/(11!)^2 + z^6/(16!)^2 + z^7/(22!)^2 + ...

what I call "tommy's hypergeo function".

Clearly t(z) is an entire function.

Again we have the property

tn(z) = 0  ==> z is a negative real !

t(z) = 0  ==> z is a negative real !

These coefficients are also sharp in many ways.

The pattern is 2,4,7,11,16,22

2 + 2 = 4

4 + 3 = 7

7 + 4 = 11

11 + 5 = 16

16 + 6 = 22

etc

so closely related to the triangular numbers + 1.

These polynomials (fn and tn) are truncated taylors but many terms can be removed and we still get the same property.

How do these functions behave ?

Due to their fast convergeance many methods and theorems for polynomials carry over to these entire functions.

More examples of similar function with the same properties 

A(z) = 1 + z  + z^2/3! + z^3/6! + z^4/10! + z^5/15! + z^6/21! + z^7/28! + ...

B(z) = 1 + z + z^2/4 + z^3/4^3 + z^4/4^6 + z^5/4^10 + z^6/4^15 + z^7/4^21 + ...

Again the triangular numbers show up.

More research is required.

The growth rate of these functions are also interesting.

Consider the function C(z) such that

C(0) = 1

dC/dz C(z) = C(z/4)

Then 

C(z) =   1 + z + z^2/(4 * 2!) + z^3/(4^3 * 3!) + z^4/(4^6 * 4!) + z^5/(4^10 * 5!) + z^6/(4^15 * 6!) + z^7/(4^21 * 7!) + ...

C(z) is the so-called Borel transform of B(z) or in other words the coefficients are multiplied by n!.

C(x) and B(x) are very brute estimated asymptotic to around exp(ln(x)^c) for real positive x being large and c in the interval [3/2 , 2].  

So does C(z) also satisfy this property of only having real zero's ?

Yes C(z) and Cn(z) also satisfy the property. 

Strongly related : the page in the menu " Fake function theory " 

Thank you for your intrest.

tommy1729