[seqfan] Constant Arising from a Certain Functional Eq

[Paul D Hanna]

SeqFans, 
    Consider the functional equation: 
  
(1) F(x,t) = exp(x) + t*x^2*F'(x,t)/F(x,t) 
 
where the logarithmic derivative is taken wrt x (not t). 
 
Clearly, a solution F(x,t) to (1) as a series in x is determined by the parameter t. 
For example, 
F(x,1) = 1 + x + 3*x^2/2! + 13*x^3/3! + 73*x^4/4! + 481*x^5/5! + 3001*x^6/6! - 8819*x^7/7! - 1883951*x^8/8! +...
 
F(x,2) = 1 + x + 5*x^2/2! + 49*x^3/3! + 865*x^4/4! + 25921*x^5/5! + 1236001*x^6/6! + 87338161*x^7/7! +...
 
Observe that in F(x,1) negative coefficients appear, 
while F(x,2) consists of only positive coefficients. 
 
In fact, some coefficients in F(x,t) are negative for any t<=1, 
and F(x,t) consists entirely of positive coefficients when t>=2. 
  
 
So then, what is the least t such that F(x,t) consists entirely of positive coefficients? 
 
 
It is interesting that such a threshhold value of t exists, and equals (approx): 
 
1.029360728932960666742865617647360272428697824009903266405204289055749580312045961621266246011201846...
 
Can this constant be expressed in terms of other known constants? 
 
  
This constant is obtained by regarding the coefficients C(n,t) of x^n in F(x,t) as polynomials in t: 
 
   F(x,t) = Sum_{n>=0} C(n,t)*x^n/n!  
 
and then finding the limit of the roots of the polynomials that are found in a suitable interval. 
That is, let r(n) be within the interval (1,2) and satisfy 
   C(n,r(n)) = 0,  
then the above constant equals limit r(n). 
(Here we determined the interval (1,2) from our observation of negative and positive coefficients; 
more specific conditions on the roots are needed for the general case.) 
  
The coefficient polynomials begin:
C(0,t) = 1;
C(1,t) = 1;
C(2,t) = 2*t + 1;
C(3,t) = 12*t^2 + 1;
C(4,t) = 144*t^3 - 72*t^2 + 1;
C(5,t) = 2880*t^4 - 2640*t^3 + 240*t^2 + 1;
C(6,t) = 86400*t^5 - 108000*t^4 + 25200*t^3 - 600*t^2 + 1;
C(7,t) = 3628800*t^6 - 5503680*t^5 + 2036160*t^4 - 171360*t^3 + 1260*t^2 + 1;
C(8,t) = 203212800*t^7 - 352235520*t^6 + 172730880*t^5 - 26530560*t^4 + 940800*t^3 - 2352*t^2 + 1;
C(9,t) = 14631321600*t^8 - 27991111680*t^7 + 16634419200*t^6 - 3674522880*t^5 + 272160000*t^4 - 4451328*t^3 + 4032*t^2 + 1;
C(10,t) = 1316818944000*t^9 - 2719858176000*t^8 + 1860093849600*t^7 - 525486528000*t^6 + 60495724800*t^5 - 2356361280*t^4 + 18869760*t^3 - 6480*t^2 + 1;
...
>From this we see that the real roots in the interval (1,2) are already converging: 
r(7) = 1.004549380068308349645605642...
r(8) = 1.022716018771956066492600608...
r(9) = 1.027860824033290897499412851...
r(10) = 1.029071227426111538733060520...
...
  
It may be interesting to study variants of (1) to arrive at other constants. 
 
Thanks, 
   Paul