[seqfan] Functional Equation Challenge

Wed Jun 8 05:20:21 CEST 2016

SeqFans, 
       Here is a challenge for you, the solution being as yet unknown to me, and yet does not seem to be intractable. 
And there are sequences of coefficients for certain A(x) that may be appropriate for submission to the OEIS. 
    OBJECTIVE.   
Given F(x) with F(0)=1, suppose A(x) satisfies 
    A( +A(x)^2 * F(A(x)) ) = x^2 
then find G(x) such that 
    A( -A(x)^2 * F(A(x)) ) = G(-x^2). 

OBSERVATION: 
For the same A(x), F(x), and G(x), we have 
    A( +sqrt( A(x^2 * F(x)) ) ) = x 
and 
    A( -sqrt( A(x^2 * F(x)) ) ) = G(-x). 

Below I give 6 simple examples. 

Note that we may generate A(x) for a given F(x) by iterating the relation 
   A(x) = Series_Reversion( sqrt( A(x^2 * F(x)) ) ).   

How does one determine G(x) from F(x)? 

Solution, anyone? 
      Paul 

EXAMPLES. 

(1) F(x) = 1 - x. 
If 
 A( A^2 - A^3 ) = x^2 
then 
 A( A^3 - A^2 ) = (1-x^2 - sqrt(1 + 2*x^2 + 3*x^4))/2. 

Equivalently, if 
 A( +sqrt( A(x^2 - x^3) ) ) = x 
then 
 A( -sqrt( A(x^2 - x^3) ) ) = (1-x - sqrt(1 + 2*x + 3*x^2))/2. 

Series for A(x) begins: 
A(x) = x + 1/2*x^2 + 3/8*x^3 + 3/4*x^4 + 175/128*x^5 + 41/16*x^6 + 4947/1024*x^7 + 321/32*x^8 + 687611/32768*x^9 + 11403/256*x^10 + 25132181/262144*x^11 + 107305/512*x^12 + 1941554203/4194304*x^13 + 2111325/2048*x^14 + 77643067507/33554432*x^15 + 21427329/4096*x^16 + 25549683166419/2147483648*x^17 + 1782548851/65536*x^18 + 1073363084982753/17179869184*x^19 + 18891311061/131072*x^20 + 91744420207896017/274877906944*x^21 + 406630578535/524288*x^22 + 3975787925128277349/2199023255552*x^23 + 4432136534071/1048576*x^24 +...

This A(x) is the g.f. of sequence https://oeis.org/A273925 

(2) F(x) = (1 - x)^2. 
If 
 A( +A^2*(1 - A)^2 ) = x^2 
then 
 A( -A^2*(1 - A)^2 ) = (1 - sqrt(1 + 4*x^2 - 4*x^4))/2. 

Equivalently, if 
 A( +sqrt( A(x^2*(1-x)^2) ) ) = x 
then 
 A( -sqrt( A(x^2*(1-x)^2) ) ) = (1 - sqrt(1 + 4*x - 4*x^2))/2.

Series for A(x) begins: 
A(x) = x + x^2 + 3/2*x^3 + 4*x^4 + 89/8*x^5 + 65/2*x^6 + 1577/16*x^7 + 623/2*x^8 + 128939/128*x^9 + 26537/8*x^10 + 2839981/256*x^11 + 300679/8*x^12 + 131796541/1024*x^13 + 7116451/16*x^14 + 3172671361/2048*x^15 + 86923531/16*x^16 + 628050354867/32768*x^17 + 8701335259/128*x^18 + 15875990451025/65536*x^19 + 110967837005/128*x^20 + 816419638391623/262144*x^21 + 2874116231667/256*x^22 + 21286039674082703/524288*x^23 + 37694485633023/256*x^24 +...

(3) F(x) = 1/(1 + x). 
If 
 A( +A^2/(1 + A) ) = x^2 
then 
 A( -A^2/(1 + A) ) = -x^2/(1 + x^2). 

Equivalently, if 
 A( +sqrt( A(x^2/(1+x)) ) ) = x 
then 
 A( -sqrt( A(x^2/(1+x)) ) ) = -x/(1 + x).

Series for A(x) begins: 
A(x) = x + 1/2*x^2 - 1/8*x^3 - 1/4*x^4 + 23/128*x^5 + 5/16*x^6 - 137/1024*x^7 - 11/32*x^8 + 1355/32768*x^9 + 83/256*x^10 + 12265/262144*x^11 - 143/512*x^12 - 730253/4194304*x^13 + 337/2048*x^14 + 11076263/33554432*x^15 + 173/4096*x^16 - 952381069/2147483648*x^17 - 19837/65536*x^18 + 7666365205/17179869184*x^19 + 71845/131072*x^20 - 62235568055/274877906944*x^21 - 329237/524288*x^22 - 785600572735/2199023255552*x^23 + 297635/1048576*x^24 +...

(4) F(x) = 1/(1 + x)^2. 
If 
 A( +A^2/(1 + A)^2 ) = x^2 
then 
 A( -A^2/(1 + A)^2 ) = -x^2/(1 + 2*x^2). 

Equivalently, if 
 A( +sqrt( A(x^2/(1+x)^2) ) ) = x 
then 
 A( -sqrt( A(x^2/(1+x)^2) ) ) = -x/(1 + 2*x).

Series for A(x) begins: 
A(x) = x + x^2 + 1/2*x^3 + 1/8*x^5 + 1/2*x^6 + 3/16*x^7 - 1/2*x^8 - 37/128*x^9 + 5/8*x^10 + 143/256*x^11 - 5/8*x^12 - 939/1024*x^13 + 7/16*x^14 + 2891/2048*x^15 + 3/16*x^16 - 62701/32768*x^17 - 189/128*x^18 + 135707/65536*x^19 + 437/128*x^20 - 294785/262144*x^21 - 1301/256*x^22 - 1170203/524288*x^23 + 835/256*x^24 +...

(5) F(x) = 1/(1 + x)^3. 
If 
 A( +A^2/(1 + A)^3 ) = x^2 
then 
 A( -A^2/(1 + A)^3 ) = (1 - sqrt(1 + 4*x^2))^3 / (8*x^4). 

Equivalently, if 
 A( +sqrt( A(x^2/(1+x)^3) ) ) = x 
then 
 A( -sqrt( A(x^2/(1+x)^3) ) ) = (1 - sqrt(1 + 4*x))^3 / (8*x^2). 

Series for A(x) begins: 
A(x) = x + 3/2*x^2 + 15/8*x^3 + 11/4*x^4 + 663/128*x^5 + 159/16*x^6 + 18375/1024*x^7 + 1077/32*x^8 + 2218443/32768*x^9 + 35129/256*x^10 + 72010329/262144*x^11 + 285225/512*x^12 + 4837378931/4194304*x^13 + 4921779/2048*x^14 + 168174620151/33554432*x^15 + 43083085/4096*x^16 + 47783653326963/2147483648*x^17 + 3098129673/65536*x^18 + 1729968190330309/17179869184*x^19 + 28222972077/131072*x^20 + 127092144961938633/274877906944*x^21 + 522055584449/524288*x^22 + 4726720740427850481/2199023255552*x^23 + 4878182483859/1048576*x^24 +...

(6) F(x) = 1/(1 + x)^4. 
If 
 A( +A^2/(1 + A)^4 ) = x^2 
then 
 A( -A^2/(1 + A)^4 ) = (1+x^2 - sqrt(1 + 6*x^2 + x^4))^2 / (4*x^2). 

Equivalently, if 
 A( +sqrt( A(x^2/(1+x)^4) ) ) = x 
then 
 A( -sqrt( A(x^2/(1+x)^4) ) ) = (1+x - sqrt(1 + 6*x + x^2))^2 / (4*x). 

Series for A(x) begins: 
A(x) = x + 2*x^2 + 4*x^3 + 10*x^4 + 57/2*x^5 + 84*x^6 + 253*x^7 + 786*x^8 + 20005/8*x^9 + 8075*x^10 + 52801/2*x^11 + 87300*x^12 + 4663327/16*x^13 + 980840*x^14 + 26589951/8*x^15 + 11332300*x^16 + 4972420403/128*x^17 + 535230605/4*x^18 + 14812211851/32*x^19 + 3214929249/2*x^20 + 1434089268063/256*x^21 + 39169345807/2*x^22 + 8789660232771/128*x^23 + 482835241865/2*x^24 +...

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