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Rational Solutions of a Nonlinear Functional Equation Related to Mahler's Equation
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
199, 489]494 Ž1996.
0156
Rational Solutions of a Nonlinear Functional Equation Related to Mahler’s Equation Charles Pegis 229 Green Oaks Dri¨ e, Ri¨ erside, California 92507 Submitted by William F. Ames Received May 15, 1995
The purpose of this paper is to prove the following two related theorems. THEOREM 1. The only nonconstant rational solutions of the functional equation F Ž z 2 . s AF Ž z . q B q CrF Ž z .
Ž 1.
Ž A, B, C constants . are: A s C s 1r2;
B s 0;
A s C s 1r4; A s C s y1r2;
F Ž z . s Ž 1 q z . rŽ 1 y z . 2
B s 1r2; B s 2;
F Ž z . s Ž 1 q z . rŽ 1 y z .
Ž 2. 2
Ž 3.
F Ž z . s Ž 1 y z q z 2 . rŽ 1 q z q z 2 .
Ž 4. and solutions obtained from Ž2., Ž3., and Ž4. by either of the transformations F Ž z . ª kF Ž z . ;
A ª A; zªz
B ª kB;
C ª k 2 C,
m
Ž 5. Ž 6.
where k is an arbitrary constant and m is a positi¨ e or negati¨ e integer. THEOREM 2. The only nonconstant rational solutions of Mahler’s functional equation w1x, GŽ z 2 . s G2 Ž z . q l,
Ž 7.
489 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
490
CHARLES PEGIS
are G Ž z . s zy1 ,
l s 0; l s y2;
Ž 8.
G Ž z . s z q 1rz,
Ž 9.
and solutions obtained from Ž8. and Ž9. by the transformation Ž6.. E¨ idently both Ž5. and Ž6. transform Ž1. into an equation of the same form, while only Ž6. does so to Ž7.. Both Ž5. and Ž6. transform rational solutions into rational solutions. It will be seen below that Ž7. may be transformed into Ž1. by a bilinear transformation, so that Theorem 2 follows immediately from Theorem 1. Equation Ž7. was studied in w1x by Mahler, who did not, however, consider the question of rational solutions. The proof of Theorem 1 is as follows. Substitution of z and yz in Ž1. and subtraction gives A F Ž z . y F Ž yz . q C 1rF Ž z . y 1rF Ž yz . s 0
Ž 10 .
or F Ž z . y F Ž yz .
A y CrF Ž z . F Ž yz . s 0.
Ž 11 .
Now if F Ž z . is nonconstant and rational, there is no loss of generality in assuming that F Ž z . is not an even function of z. For if it is even, then every exponent of z occurring is an even integer. Let M be the highest power of 2 which divides all of these exponents. Then F Ž z . is obtainable by the transformation of type Ž6., z ª z M,
Ž 12 .
from a rational function of z which is not even. If F Ž z . is not even, it follows from Ž11. that A / 0;
C / 0;
F Ž z . F Ž yz . s CrA
Ž 13 .
and it may be ensured by a transformation of type Ž5. that C s A;
F Ž 0 . s Ž CrA .
1r2
s1
Ž 14 .
from which, by Ž13., F Ž z . F Ž yz . s 1
Ž 15 .
B s Ž 1 y A . F Ž 0 . y CrF Ž 0 . s 1 y 2 A.
Ž 16 .
and, by Ž1.,
491
NONLINEAR FUNCTIONAL EQUATION
Since F Ž z . is a rational function, F Ž z . s P Ž z . rQ Ž z . ,
Ž 17.
where P Ž z ., QŽ z . are polynomials. It may be assumed that Ž17. is in its lowest terms and, by Ž15., that P Ž0. s QŽ0. s 1. Hence, from Ž15., P Ž z . rQ Ž z . s Q Ž yz . rP Ž yz . ,
Ž 18 .
so that Q Ž z . s P Ž yz . ;
F Ž z . s P Ž z . rP Ž yz . .
Ž 19 .
Substituting Ž14., Ž16., and Ž19. in Ž1. gives PŽ z2 . P Ž yz 2 .
s
A P 2 Ž z . q P 2 Ž yz . q Ž 1 y 2 A . P Ž z . P Ž yz . P Ž z . P Ž yz .
, Ž 20 .
where each fraction is in its lowest terms and P Ž0. s 1. Hence, P Ž z 2 . s A P 2 Ž z . q P 2 Ž yz . q Ž 1 y 2 A . P Ž z . P Ž yz .
Ž 21 .
P Ž yz 2 . s P Ž z . P Ž yz . .
Ž 22 .
Using Ž22., Eq. Ž21. may be written P Ž z 2 . s A P Ž z . y P Ž yz .
2
q P Ž z . P Ž yz .
Ž 23 .
or R Ž z 2 . s AR 2 Ž z . ,
Ž 24 .
R Ž z . s P Ž z . y P Ž yz . .
Ž 25 .
where
The general solution of Ž24. may be obtained by the substitutions w s log Ž log z . rlog 2
Ž 26 .
W Ž w . s log R Ž z .
Ž 27 .
W Ž w q 1 . s 2W Ž w . q log A
Ž 28 .
from which Ž24. becomes
which is a linear difference equation with constant coefficients whose general solution is W Ž w . s ylog A q 2 wf Ž w . , where f Ž w . is an arbitrary unit periodic function of w.
Ž 29 .
492
CHARLES PEGIS
From Ž29., R Ž z . s Ay1 z f Ž w . ,
Ž 30 .
but by Ž25. RŽ z . is an odd polynomial, so that
f Ž w. s 2m q 1
Ž 31 .
for some nonnegative integer m. It follows from Ž25., Ž30., and Ž31. that P Ž z . s S Ž z . q Ay1 z 2 mq1r2,
Ž 32 .
where SŽ z . is an even polynomial. Substituting Ž32. in Ž22. therefore gives S Ž z 2 . s S 2 Ž z . q az 4 mq2 ,
Ž 33 .
a s Ay1 r2 y Ay2 r4.
Ž 34 .
where
Since S Ž z . is an even polynomial, it may be written S Ž z . s s Ž z 4 . q z 2t Ž z 4 . ,
Ž 35 .
where s Ž z . and t Ž z . are polynomials and, since P Ž0. s 1,
s Ž 0 . s 1.
Ž 36 .
Substituting Ž35. in Ž33. gives
s Ž z 8 . q z 4t Ž z 8 . s s 2 Ž z 4 . q 2 z 2s Ž z 4 . t Ž z 4 . q z 4t 2 Ž z 4 . q az 4 mq2 Ž 37 . and equating the sums of terms with exponents ' 0 and ' 2 Žmod 4. on both sides of Ž37. gives
s Ž z 8 . q z 4t Ž z 8 . s s 2 Ž z 4 . q z 4t 2 Ž z 4 .
Ž 38 .
0 s 2 z 2s Ž z 4 . t Ž z 4 . q az 4 mq2 .
Ž 39 .
The only polynomial solutions of Ž39. satisfying Ž36. are
s Ž z4 . s 1
Ž 40 .
t Ž z 4 . s yaz 4 m r2
Ž 41 .
493
NONLINEAR FUNCTIONAL EQUATION
and substituting Ž40. and Ž41. in Ž38. gives 1 y az 8 mq4r2 s 1 q a2 z 8 mq4r4
Ž 42 .
ar2 q a2r4 s 0.
Ž 43 .
or
Substituting Ž34. in Ž43. gives
Ž Ay1r2 y Ay2r4 . r2 q Ž Ay1r2 y Ay2r4.
2
r4 s 0
Ž 44 .
of which the roots are A s 1r2;
A s 1r4;
A s y1r2
Ž 45 .
and, finally, substituting Ž45., Ž41., Ž40., Ž35., Ž34., and Ž32. in Ž19. gives the nonconstant rational solutions of Eq. Ž1., A s C s 1r2; A s C s 1r4; A s C s y1r2;
B s 0; B s 1r2;
B s 2;
F Ž z . s Ž 1 q z n . rŽ 1 y z n . 2
F Ž z . s Ž 1 q z n . rŽ 1 y z n .
Ž 46 . 2
Ž 47 .
F Ž z . s Ž 1 y z q z . rŽ 1 q z q z 2 n . , Ž 48 . n
2n
n
where n s 2 m q 1.
Ž 49 .
If m s 0, these solutions are identical to Ž2., Ž3., and Ž4.. Otherwise, they are obtainable from the latter by a transformation of type Ž6.. This completes the proof of Theorem 1. Theorem 2 is now proved at once by applying the transformations GŽ z . s a Ž 1 q F Ž z . . rŽ 1 y F Ž z . .
Ž 50 .
l s ya y a 2
Ž 51 .
to Mahler’s equation Ž7.. This gives F Ž z 2 . s Ž yF Ž z . q 2 Ž 1 q a . y 1rF Ž z . . r Ž 2 a .
Ž 52 .
which is identical to Eq. Ž1. with A s C s y1r Ž 2 a . ;
B s 1 q 1ra s 1 y 2 A,
Ž 53 .
so that substituting Ž2., Ž3., and Ž4. in Ž50. and Ž53. gives the possible
494
CHARLES PEGIS
nonconstant rational solutions of Ž7.: A s 1r2; A s 1r4; A s y1r2;
a s y1;
l s 0;
G Ž z . s zy1
Ž 54 .
a s y2;
l s y2;
G Ž z . s z q zy1
Ž 55 .
a s 1;
l s y2;
G Ž z . s z q zy1 .
Ž 56 .
This completes the proof of Theorem 2.
REFERENCES 1. K. Mahler, On a special nonlinear functional equation, Proc. Roy. Soc. London Ser. A 378 Ž1981., 155]178.
199, 489]494 Ž1996.
0156
Rational Solutions of a Nonlinear Functional Equation Related to Mahler’s Equation Charles Pegis 229 Green Oaks Dri¨ e, Ri¨ erside, California 92507 Submitted by William F. Ames Received May 15, 1995
The purpose of this paper is to prove the following two related theorems. THEOREM 1. The only nonconstant rational solutions of the functional equation F Ž z 2 . s AF Ž z . q B q CrF Ž z .
Ž 1.
Ž A, B, C constants . are: A s C s 1r2;
B s 0;
A s C s 1r4; A s C s y1r2;
F Ž z . s Ž 1 q z . rŽ 1 y z . 2
B s 1r2; B s 2;
F Ž z . s Ž 1 q z . rŽ 1 y z .
Ž 2. 2
Ž 3.
F Ž z . s Ž 1 y z q z 2 . rŽ 1 q z q z 2 .
Ž 4. and solutions obtained from Ž2., Ž3., and Ž4. by either of the transformations F Ž z . ª kF Ž z . ;
A ª A; zªz
B ª kB;
C ª k 2 C,
m
Ž 5. Ž 6.
where k is an arbitrary constant and m is a positi¨ e or negati¨ e integer. THEOREM 2. The only nonconstant rational solutions of Mahler’s functional equation w1x, GŽ z 2 . s G2 Ž z . q l,
Ž 7.
489 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
490
CHARLES PEGIS
are G Ž z . s zy1 ,
l s 0; l s y2;
Ž 8.
G Ž z . s z q 1rz,
Ž 9.
and solutions obtained from Ž8. and Ž9. by the transformation Ž6.. E¨ idently both Ž5. and Ž6. transform Ž1. into an equation of the same form, while only Ž6. does so to Ž7.. Both Ž5. and Ž6. transform rational solutions into rational solutions. It will be seen below that Ž7. may be transformed into Ž1. by a bilinear transformation, so that Theorem 2 follows immediately from Theorem 1. Equation Ž7. was studied in w1x by Mahler, who did not, however, consider the question of rational solutions. The proof of Theorem 1 is as follows. Substitution of z and yz in Ž1. and subtraction gives A F Ž z . y F Ž yz . q C 1rF Ž z . y 1rF Ž yz . s 0
Ž 10 .
or F Ž z . y F Ž yz .
A y CrF Ž z . F Ž yz . s 0.
Ž 11 .
Now if F Ž z . is nonconstant and rational, there is no loss of generality in assuming that F Ž z . is not an even function of z. For if it is even, then every exponent of z occurring is an even integer. Let M be the highest power of 2 which divides all of these exponents. Then F Ž z . is obtainable by the transformation of type Ž6., z ª z M,
Ž 12 .
from a rational function of z which is not even. If F Ž z . is not even, it follows from Ž11. that A / 0;
C / 0;
F Ž z . F Ž yz . s CrA
Ž 13 .
and it may be ensured by a transformation of type Ž5. that C s A;
F Ž 0 . s Ž CrA .
1r2
s1
Ž 14 .
from which, by Ž13., F Ž z . F Ž yz . s 1
Ž 15 .
B s Ž 1 y A . F Ž 0 . y CrF Ž 0 . s 1 y 2 A.
Ž 16 .
and, by Ž1.,
491
NONLINEAR FUNCTIONAL EQUATION
Since F Ž z . is a rational function, F Ž z . s P Ž z . rQ Ž z . ,
Ž 17.
where P Ž z ., QŽ z . are polynomials. It may be assumed that Ž17. is in its lowest terms and, by Ž15., that P Ž0. s QŽ0. s 1. Hence, from Ž15., P Ž z . rQ Ž z . s Q Ž yz . rP Ž yz . ,
Ž 18 .
so that Q Ž z . s P Ž yz . ;
F Ž z . s P Ž z . rP Ž yz . .
Ž 19 .
Substituting Ž14., Ž16., and Ž19. in Ž1. gives PŽ z2 . P Ž yz 2 .
s
A P 2 Ž z . q P 2 Ž yz . q Ž 1 y 2 A . P Ž z . P Ž yz . P Ž z . P Ž yz .
, Ž 20 .
where each fraction is in its lowest terms and P Ž0. s 1. Hence, P Ž z 2 . s A P 2 Ž z . q P 2 Ž yz . q Ž 1 y 2 A . P Ž z . P Ž yz .
Ž 21 .
P Ž yz 2 . s P Ž z . P Ž yz . .
Ž 22 .
Using Ž22., Eq. Ž21. may be written P Ž z 2 . s A P Ž z . y P Ž yz .
2
q P Ž z . P Ž yz .
Ž 23 .
or R Ž z 2 . s AR 2 Ž z . ,
Ž 24 .
R Ž z . s P Ž z . y P Ž yz . .
Ž 25 .
where
The general solution of Ž24. may be obtained by the substitutions w s log Ž log z . rlog 2
Ž 26 .
W Ž w . s log R Ž z .
Ž 27 .
W Ž w q 1 . s 2W Ž w . q log A
Ž 28 .
from which Ž24. becomes
which is a linear difference equation with constant coefficients whose general solution is W Ž w . s ylog A q 2 wf Ž w . , where f Ž w . is an arbitrary unit periodic function of w.
Ž 29 .
492
CHARLES PEGIS
From Ž29., R Ž z . s Ay1 z f Ž w . ,
Ž 30 .
but by Ž25. RŽ z . is an odd polynomial, so that
f Ž w. s 2m q 1
Ž 31 .
for some nonnegative integer m. It follows from Ž25., Ž30., and Ž31. that P Ž z . s S Ž z . q Ay1 z 2 mq1r2,
Ž 32 .
where SŽ z . is an even polynomial. Substituting Ž32. in Ž22. therefore gives S Ž z 2 . s S 2 Ž z . q az 4 mq2 ,
Ž 33 .
a s Ay1 r2 y Ay2 r4.
Ž 34 .
where
Since S Ž z . is an even polynomial, it may be written S Ž z . s s Ž z 4 . q z 2t Ž z 4 . ,
Ž 35 .
where s Ž z . and t Ž z . are polynomials and, since P Ž0. s 1,
s Ž 0 . s 1.
Ž 36 .
Substituting Ž35. in Ž33. gives
s Ž z 8 . q z 4t Ž z 8 . s s 2 Ž z 4 . q 2 z 2s Ž z 4 . t Ž z 4 . q z 4t 2 Ž z 4 . q az 4 mq2 Ž 37 . and equating the sums of terms with exponents ' 0 and ' 2 Žmod 4. on both sides of Ž37. gives
s Ž z 8 . q z 4t Ž z 8 . s s 2 Ž z 4 . q z 4t 2 Ž z 4 .
Ž 38 .
0 s 2 z 2s Ž z 4 . t Ž z 4 . q az 4 mq2 .
Ž 39 .
The only polynomial solutions of Ž39. satisfying Ž36. are
s Ž z4 . s 1
Ž 40 .
t Ž z 4 . s yaz 4 m r2
Ž 41 .
493
NONLINEAR FUNCTIONAL EQUATION
and substituting Ž40. and Ž41. in Ž38. gives 1 y az 8 mq4r2 s 1 q a2 z 8 mq4r4
Ž 42 .
ar2 q a2r4 s 0.
Ž 43 .
or
Substituting Ž34. in Ž43. gives
Ž Ay1r2 y Ay2r4 . r2 q Ž Ay1r2 y Ay2r4.
2
r4 s 0
Ž 44 .
of which the roots are A s 1r2;
A s 1r4;
A s y1r2
Ž 45 .
and, finally, substituting Ž45., Ž41., Ž40., Ž35., Ž34., and Ž32. in Ž19. gives the nonconstant rational solutions of Eq. Ž1., A s C s 1r2; A s C s 1r4; A s C s y1r2;
B s 0; B s 1r2;
B s 2;
F Ž z . s Ž 1 q z n . rŽ 1 y z n . 2
F Ž z . s Ž 1 q z n . rŽ 1 y z n .
Ž 46 . 2
Ž 47 .
F Ž z . s Ž 1 y z q z . rŽ 1 q z q z 2 n . , Ž 48 . n
2n
n
where n s 2 m q 1.
Ž 49 .
If m s 0, these solutions are identical to Ž2., Ž3., and Ž4.. Otherwise, they are obtainable from the latter by a transformation of type Ž6.. This completes the proof of Theorem 1. Theorem 2 is now proved at once by applying the transformations GŽ z . s a Ž 1 q F Ž z . . rŽ 1 y F Ž z . .
Ž 50 .
l s ya y a 2
Ž 51 .
to Mahler’s equation Ž7.. This gives F Ž z 2 . s Ž yF Ž z . q 2 Ž 1 q a . y 1rF Ž z . . r Ž 2 a .
Ž 52 .
which is identical to Eq. Ž1. with A s C s y1r Ž 2 a . ;
B s 1 q 1ra s 1 y 2 A,
Ž 53 .
so that substituting Ž2., Ž3., and Ž4. in Ž50. and Ž53. gives the possible
494
CHARLES PEGIS
nonconstant rational solutions of Ž7.: A s 1r2; A s 1r4; A s y1r2;
a s y1;
l s 0;
G Ž z . s zy1
Ž 54 .
a s y2;
l s y2;
G Ž z . s z q zy1
Ž 55 .
a s 1;
l s y2;
G Ž z . s z q zy1 .
Ž 56 .
This completes the proof of Theorem 2.
REFERENCES 1. K. Mahler, On a special nonlinear functional equation, Proc. Roy. Soc. London Ser. A 378 Ž1981., 155]178.