Continued Fractions and Generalizations with Many Limits: A Survey
aa r X i v : . [ m a t h . N T ] J a n CONTINUED FRACTIONS AND GENERALIZATIONS WITH MANYLIMITS: A SURVEY.
DOUGLAS BOWMAN AND JAMES MC LAUGHLIN
Abstract.
There are infinite processes (matrix products, continued fractions, ( r, s )-matrixcontinued fractions, recurrence sequences) which, under certain circumstances, do not con-verge but instead diverge in a very predictable way.We give a survey of results in this area, focusing on recent results of the authors. Introduction
Consider the following recurrence: x n +1 = 43 − x n . Taking 1 / ∞ to be 0 and vice versa, then regardless of the initial (real) value of this sequence,it is an interesting fact that the sequence is dense in R . The proof is illuminating.Take x = 4 / x n as n ’th approximant of the continued fraction:4 / − / − / − / − / − · · · . (1)Then, from the standard theorem on the recurrence for convergents of a continued fraction,the n ’th numerator and denominator convergents of this continued fraction, A n and B n respectively, must both satisfy the linear recurrence relation Y n = 43 Y n − − Y n − , but with different initial conditions.Now, the characteristic roots of this equation are α = 2 / i √ /
3, and β = 2 / − i √ / x n is x n = A n B n = aα n + bβ n cα n + dβ n = aλ n + bcλ n + d , where a , b , c , and d are some complex constants and λ = α/β . Notice that λ is a numberon the unit circle and is not a root of unity, so that λ n is dense on the unit circle. Theconclusion follows by noting that the linear fractional transformation z az + bcz + d must take the unit circle to R , since the values of the sequence x n are real. Date : July 31, 2006.The first author’s research was partially supported by NSF grant DMS-0300126.
After seeing this argument, one is tempted to write down the amusing identity R = 4 / − / − / − / − / − · · · . This identity is true so long as one interprets the value of the continued fraction to be theset of limits of subsequences of its sequence of approximants.Another motivating example of our work is the following theorem, one of the oldest in theanalytic theory of continued fractions [6]:
Theorem 1.1. (Stern-Stolz) Let the sequence { b n } satisfy P | b n | < ∞ . Then b + K ∞ n =1 b n diverges. In fact, for p = 0 , , lim n →∞ P n + p = A p = ∞ , lim n →∞ Q n + p = B p = ∞ , and A B − A B = 1 . The Stern-Stolz theorem gives a general class of continued fractions each of which tend totwo different limits, respectively A /B , and A /B . Here and throughout we assume thelimits for continued fractions are in b C . This makes sense because continued fractions can beviewed as the composition of linear fractional transformations and such functions have b C astheir natural domain and codomain.Before leaving the Stern-Stolz theorem, we wish to remark that although the theorem isusually termed a “divergence theorem”, this terminology is a bit misleading; the theoremactually shows that although the continued fractions of this form diverge, they do so bytending to two limits in a precisely controlled way. In this paper we study extensions ofthis phenomenon and investigate just how far one can go in this direction. Thus, althoughthroughout this paper we refer to certain of our results as “divergence” theorems, most ofthem actually give explicit results about convergent subsequences.A special case of the Stern-Stolz theorem gives a result on the famous Rogers-Ramanujancontinued fraction: 1 + q q q q · · · , (2)The Stern-Stolz theorem gives that for | q | > /q + 11 /q + 11 /q + 11 /q · · · + 11 /q n + 11 /q n · · · . The Stern-Stolz theorem, however does not apply to the following continued fraction givenby Ramanujan: −
11 + q + −
11 + q + −
11 + q + · · · . (3)Recently in [1] Andrews, Berndt, et al. proved a claim made by Ramanujan in his lostnotebook ([9], p.45) about (3). To describe Ramanujan’s claim, we first need some notation. ONTINUED FRACTIONS WITH MANY LIMITS 3
Throughout take q ∈ C with | q | <
1. The following standard notation for q -products willalso be employed:( a ) := ( a ; q ) := 1 , ( a ) n := ( a ; q ) n := n − Y k =0 (1 − a q k ) , if n ≥ , and ( a ; q ) ∞ := ∞ Y k =0 (1 − a q k ) , | q | < . Set ω = e πi/ . Ramanujan’s claim was that, for | q | < lim n →∞ (cid:18) −
11 + q −
11 + q − · · · −
11 + q n + a (cid:19) = − ω (cid:18) Ω − ω n +1 Ω − ω n − (cid:19) . ( q ; q ) ∞ ( q ; q ) ∞ , (4) where Ω := 1 − aω − aω ( ω q, q ) ∞ ( ωq, q ) ∞ . Ramanujan’s notation is confusing, but what his claim means is that the limit exists as n → ∞ in each of the three congruence classes modulo 3, and that the limit is given by theexpression on the right side of (4). Also, the appearance of the variable a in this formula isa bit of a red herring; from elementary properties of continued fractions, one can derive theresult for general a from the a = 0 case.Now (1) is different from the other examples in that it has subsequences of approximantstending to infinitely many limits. Nevertheless, all of the examples above, including (1), arespecial cases of a general result on continued fractions (Theorem 4.1 below). To deal withboth of these situations we introduce the notion of the limit set of a sequence.The limit set of the sequence is defined to be the set of all limits of convergent subsequences.Limit sets should not be confused with sets of limit points. Thus, for example, the sequence { , , , . . . } has limit set { } although the set of limit (accumulation) points of the setof values of the sequence is empty. Limit sets need to be introduced so that sequenceswith constant subsequences will have the values of these subsequences included among thepossible limits. Certain periodic continued fractions have this property. To avoid confusionwe designate the limit set of a sequence { s n } n ≥ by l.s. ( s n ).Our initial research [2] dealt with cases in which the limit set was finite. In [3] we ex-tended our methods to give a uniform treatment of finite and infinite cases. In fact, in [3],we studied asymptotics for approximants for infinite matrix products, continued fractions,and recurrence relations of Poincar´e type. Limit set information easily follows from theasymptotics.In the papers [2] and [3], the authors studied limit sets in the specific context of sequencesof the form f n Y i =1 D i ! , where D i is a sequence of complex matrices and f is a function with values in some compactmetric space. DOUGLAS BOWMAN AND JAMES MC LAUGHLIN Definitions, Notation and Terminology
Limit set equalities in this paper arise from the situationlim n →∞ d ( s n , t n ) = 0in some metric space ( X, d ). Accordingly, it makes sense to define the equivalence relation ∼ on sequences in X by { s n } ∼ { t n } ⇐⇒ lim n →∞ d ( s n , t n ) = 0. In this situation we referto sequences { s n } and { t n } as being asymptotic to each other. Abusing notation, we oftenwrite s n ∼ t n in place of { s n } ∼ { t n } . More generally, we frequently write sequences withoutbraces when it is clear from context that we are speaking of a sequence, and not the n thterm.Let M d ( C ) denote the set of d × d matrices of complex numbers topologised using the l ∞ norm, denoted by || · || . Let I denote the identity matrix. When we use product notationfor matrices, the product is taken from left to right; thus n Y i =1 A i := A A · · · A n . An infinite continued fraction K ∞ n =1 a n b n := a b + a b + a b + · · · (5)is said to converge if lim n →∞ a b + a b + a b + · · · + a n b n exists in b C . Let { ω n } be a sequence of complex numbers. Iflim n →∞ a b + a b + a b + · · · + a n b n + ω n exist, then this limit is called the modified limit of K ∞ n =1 a n /b n with respect to the sequence { ω n } . Detailed discussions of modified continued fractions as well as further pointers to theliterature are given in [6].We follow the common convention in analysis of denoting the group of points on the unitcircle by T , or by T ∞ , and its subgroup of roots of unity of order m , m finite, by T m . (Note: T ∞ often denotes the group of all roots of unity; here it denotes the whole circle group.)3. Theorems of Infinite Matrix Products
The classic theorem on the convergence of infinite products of matrices seems first to havebeen given clearly by Wedderburn [10].
Proposition 1. (Wedderburn [10, 11] ) Let A i ∈ M d ( C ) for i ≥ . Then P i ≥ || A i || < ∞ implies that Q i ≥ ( I + A i ) converges in M d ( C ) . In [2], our initial motivation was to generalize the Ramanujan continued fraction withthree limits (4) to a continued fraction with m limits, m ≥
3. This led us to consider infinitesequences of matrices converging to 2 × p × p matrices, p ≥
2, with similar properties. In [2] we proved the following result.
ONTINUED FRACTIONS WITH MANY LIMITS 5
Proposition 2.
Let p ≥ be an integer and let M be a p × p matrix that is diagonalizableand whose eigenvalues are roots of unity. Let I denote the p × p identity matrix and let m be the least positive integer such that M m = I. For a p × p matrix G , let || G || ∞ = max ≤ i,j ≤ p | G ( i,j ) | , where G ( i,j ) denotes the element of G in row i and column j . Suppose { D n } ∞ n =1 is a sequenceof matrices such that ∞ X n =1 || D n − M || ∞ < ∞ . Then F := lim k →∞ km Y n =1 D n exists. Here the matrix product means either D D . . . or . . . D D . Further, for each j , ≤ j ≤ m − , lim k →∞ km + j Y n =1 D n = M j F or F M j , depending on whether the products are taken to the left or right. A natural progression was to replace the matrix M in the proposition above with a se-quence of matrices { M i } . In [3] we proved the following result. Theorem 1.
Suppose { M i } and { D i } are sequences of complex matrices such that the twosequences (for ǫ = ± ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n Y i =1 M i ! ǫ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (6) are bounded and X i ≥ k D i − M i k < ∞ . (7) Then F := lim n →∞ n Y i =1 D i ! n Y i =1 M i ! − (8) exists and det( F ) = 0 if and only if det( D i ) = 0 for all i ≥ .As sequences n Y i =1 D i ∼ F n Y i =1 M i . (9) More generally, let f be a continuous function from the domain ( F n Y i =1 M i : n ≥ h ) ∪ [ n ≥ h ( n Y i =1 D i ) , DOUGLAS BOWMAN AND JAMES MC LAUGHLIN for some integer h ≥ , into a metric space G . Then the domain of f is compact in M d ( C ) and f ( Q ni =1 D i ) ∼ f ( F Q ni =1 M i ) . Finally l.s. n Y i =1 D i ! = l.s. F n Y i =1 M i ! , (10) and l.s. f n Y i =1 D i !! = l.s. f F n Y i =1 M i !! . (11)Theorem 1 had several interesting applications to certain classes of continued fractions,recurrence sequences, and ( r, s )-matrix continued fractions.4. Theorems on Continued Fractions
We begin by stating our general theorem on the asymptotics and limit sets of the sequenceof approximants of a class of continued fractions. The theorem shows that the limit set is acircle (or a finite subset of a circle) on the Riemann sphere. When the limit set is a circle,although the set of approximants approaches all of its points, the approximants usually donot do so in a uniform way.The following theorem concerns the continued fraction − αβ + q α + β + p + − αβ + q α + β + p + · · · + − αβ + q n α + β + p n , (12)where the sequences p n and q n are absolutely summable and the constants α = β are pointson the unit circle. Theorem 4.1.
Let { p n } n ≥ , { q n } n ≥ be complex sequences satisfying ∞ X n =1 | p n | < ∞ , ∞ X n =1 | q n | < ∞ . Let α and β satisfy | α | = | β | = 1 , α = β with the order of λ = α/β in T being m ( where m may be infinite ) . Assume that q n = αβ for any n ≥ . Put f n ( w ) := − αβ + q α + β + p + − αβ + q α + β + p + · · · + − αβ + q n α + β + p n + w , so that f n := f n (0) is the sequence of approximants of the continued fraction (12) . Then f n ∼ h ( λ n +1 ) so that l.s. ( f n ) = h ( T m ) , where h ( z ) = az + bcz + d , with the constants a, b, c, d ∈ C given by the (existent) limits a = lim n →∞ α − n ( P n − βP n − ) , (13) b = − lim n →∞ β − n ( P n − αP n − ) ,c = lim n →∞ α − n ( Q n − βQ n − ) ,d = − lim n →∞ β − n ( Q n − αQ n − ) , ONTINUED FRACTIONS WITH MANY LIMITS 7 where P n and Q n are the n th convergents of the continued fraction (12) . Moreover, det( h ) = ad − bc = ( β − α ) ∞ Y n =1 (cid:18) − q n αβ (cid:19) = 0 , (14) and the following identities involving modified versions of (12) hold in b C : h ( ∞ ) = ac (15)= lim n →∞ − αβ + q α + β + p + − αβ + q α + β + p + · · · + − αβ + q n − α + β + p n − + − αβ + q n α + p n ; h (0) = bd (16)= lim n →∞ − αβ + q α + β + p + − αβ + q α + β + p + · · · + − αβ + q n − α + β + p n − + − αβ + q n β + p n ; and for k ∈ Z , we have h ( λ k +1 ) = aλ k +1 + bcλ k +1 + d = lim n →∞ − αβ + q α + β + p + − αβ + q α + β + p + · · · + − αβ + q n α + β + p n + ω n − k , (17) where ω n = − α n − β n α n − − β n − ∈ b C , n ∈ Z . As a first application, we can get quite precise information about the divergence behaviorof limit-1 periodic continued fractions of elliptic type (see [6] for more on limit-1 periodiccontinued fractions of elliptic type).We consider the case where the continued fraction a b + a b + a b + · · · (18)is a limit 1-periodic continued fraction of elliptic type and, in addition, X n ≥ | a n − a | < ∞ , X n ≥ | b n − b | < ∞ , for some a, b ∈ C .Set d := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b + √ b + 4 a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b − √ b + 4 a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , and define α = b + √ b + 4 a d , β = b − √ b + 4 a d . Then α = β , | α | = | β | = 1. Define, for n ≥ p n and q n by a n = a + p n , b n = b + q n . DOUGLAS BOWMAN AND JAMES MC LAUGHLIN
Thus K ∞ n =1 a + q n b + p n = d K ∞ n =1 − αβ + q n /d α + β + p n /d . The second continued fraction satisfies the conditions of Theorem 4.1. Thus this theoremcan be applied to all limit 1-periodic continued fractions of elliptic type with lim n →∞ a n = a and lim n →∞ b n = b , providing P n ≥ | a n − a | < ∞ and P n ≥ | b n − b | < ∞ . Of course, it isknown that without any restrictions on how the limit periodic sequences tend to their limits,the behavior can be quit complicated, see [6].Next, we can obtain (up to a factor of ±
1) the numbers a , b , c , and d in terms of themodified continued fractions and the product for det( h ) given in Theorem 4.1. Corollary 4.2.
The linear fractional transformation h ( z ) defined in Theorem 4.1 has thefollowing expression h ( z ) = A ( C − B ) z + B ( A − C )( C − B ) z + A − C , where A = h ( ∞ ) , B = h (0) , and C = h (1) . Moreover, the constants a , b , c , and d in thetheorem have the following formulas a = sA ( C − B ) , b = sB ( A − C ) , c = s ( C − B ) , d = s ( A − C ) , where s = ± vuut ( β − α ) Q ∞ n =1 (cid:16) − q n αβ (cid:17) ( A − B )( C − A )( B − C ) . It is interesting that the linear fractional transformation which describes the limit set ofthe divergent continued fraction K ∞ n =1 − αβ + q n α + β + p n can be described completely in terms of three convergent modified continued fractions.Let T ′ denote the image of T under h , that is, the limit set of the sequence { f n } . Themain conclusion of the theorem can be expressed by the statement f n ∼ h ( λ n +1 ) , (19)where h is the linear fractional transformation in the theorem. It is well known that when λ is not a root of unity, λ n +1 is uniformly distributed on T . However, the linear fractionaltransformation h stretches and compresses arcs of the circle T , so that the distribution of h ( λ n +1 ) in arcs of T ′ is no longer uniform. Thus, although the limit set in the case where λ is not a root of unity is a circle, the concentration of approximants is not uniform aroundthe circle.Fortunately, the distribution of approximants is completely controlled by the known pa-rameters a , b , c , and d . The following corollary gives the points on the limit sets whoseneighborhood arcs have the greatest and least concentrations of approximants. (We do nottake the space here to give a precise definition of what this means; interested readers shouldconsult the author’s paper [3].) ONTINUED FRACTIONS WITH MANY LIMITS 9
Corollary 4.3.
When m = ∞ and cd = 0 , the points on a T m + bc T m + d with the highest and lowest concentrations of approximants are ac | c | + bd | d || c | + | d | and − ac | c | + bd | d |−| c | + | d | , respectively. If either c = 0 or d = 0 , then all points on the limit set have the sameconcentration. The radius of the limit set circle in C is (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α − β | c | − | d | ∞ Y n =1 (cid:18) − q n αβ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The limit set is a line in C if and only if | c | = | d | , and in this case the point of leastconcentration is ∞ . Corollary 4.4.
If the limit set of the continued fraction in (12) is a line in C , then the pointof highest concentration of approximants in the limit set is exactly h ( ∞ ) + h (0)2 , the average of the first two modifications of (12) given in Theorem 4.1. It is also possible to derive a convergent continued fractions which have the same limit asthe modified continued fractions in Theorem 4.1. These are given in the following corollary.
Corollary 4.5.
Let α , β , { p n } , { q n } , h ( z ) , be as in Theorem 4.1. Then h ( ∞ ) = − β + q + βp α + p + ( q − αβ )( q + βp )( α + p )( q + βp ) + β ( q + βp )+ K ∞ n =3 ( q n − − αβ )( q n + βp n )( q n − + βp n − )( α + p n )( q n − + βp n − ) + β ( q n + βp n ) , (20) h (0) = − α + q + αp β + p + ( q − αβ )( q + αp )( β + p )( q + αp ) + α ( q + αp )+ K ∞ n =3 ( q n − − αβ )( q n + αp n )( q n − + αp n − )( β + p n )( q n − + αp n − ) + α ( q n + αp n ) . (21) Let k ∈ Z and assume that α/β is not a root of unity. Set ω n = − α n − k − β n − k α n − k − − β n − k − , for n ≥ k ′ := max { , k + 3 } . Then h ( λ k +1 ) = − αβ + q α + β + p + · · · + − αβ + q k ′ − α + β + p k ′ − + − αβ + q k ′ α + β + p k ′ + ω k ′ + − αβ + q k ′ +1 − ω k ′ ( α + β + p k ′ +1 + ω k ′ +1 ) α + β + p k ′ +1 + ω k ′ +1 + K ∞ n = k ′ +2 c n d n , (22) where c n = ( q n − − αβ ) − αβ + q n − ω n − ( α + β + p n + ω n ) − αβ + q n − − ω n − ( α + β + p n − + ω n − ) d n = α + β + p n + ω n − ω n − − αβ + q n − ω n − ( α + β + p n + ω n ) − αβ + q n − − ω n − ( α + β + p n − + ω n − ) . Before continuing, we give an example which illustrates some of the results mentionedabove. Let | p | , | q | <
1, and define G ( p, q, α, β ) := − αβ + qα + β + p + − αβ + q α + β + p + · · · + − αβ + q n α + β + p n + · · · . For p = 0 . q = 0 . α = exp( ı √ β = exp( ı √ p n = p n , q n = q n and k = − h ( ∞ ) = 1 . . i,h (0) = 1 . . i,h (1) = − . − . i. We then apply Corollary 4.2 and compute s = 2 . . i,a = 0 . . i,b = − . − . i,c = 0 . . i,d = − . − . i. With h ( z ) := az + bcz + d , we now compare the predicted limit set h ( T ) with the sequence of approximants. Figure 1shows the first 3000 approximants of G (0 . , . , exp( ı √ , exp( ı √ h ( T ).We see that the limit set is exactly what is predicted by Theorem 4.1. The large dots showthe points of highest (top) and lowest (bottom) points of concentration of approximants, aspredicted by Corollary 4.3, namely 1 . . i and 1 . − . i . We see thatprediction and mathematical fact agree in this case also.We next consider an example where α/β is a root of unity, so that the limit set is finite.We proceed as above to compute h ( z ) (details omitted). Figure 2 shows the first 3000approximants of G (0 . , . , exp( ı √ , exp( ı ( √
11 + 2 π/ h ( T ).Figure 3 shows the image of all seventeen 17th roots of unity under h . Once again theactual limit set and the predicted limit set agree perfectly. ONTINUED FRACTIONS WITH MANY LIMITS 11 -1 1 2 3-4-3-2-1
Figure 1.
The convergence of G (0 . , . , exp( ı √ , exp( ı √ Figure 2.
The convergence of G (0 . , . , exp( ı √ , exp( ı ( √
11 + 2 π/ K ∞ n =1 − / (4 / a = − / √ / i,b = 2 / √ / i,c = 1 ,d = − . Corollary 4.4 predicts that the highest concentration of approximants occurs at ( a/c + b/d ) / − /
3. Figure 4 shows the distribution of the first 1200 approximants of the
Figure 3.
The image of the seventeen 17th roots of unity under h .continued fraction (about 100 extreme values were omitted), once again showing agreementwith the theory. -4 -2 0 2 420406080100 Figure 4.
The distribution of the first 1200 approximants of K ∞ n =1 − / (4 / An Infinite Family of Divergence Theorems.
An interesting special case of The-orem 4.1 occurs when α and β are distinct m -th roots of unity ( m ≥ − αβ + q α + β + p + − αβ + q α + β + p + − αβ + q α + β + p + − αβ + q α + β + p + · · · becomes limit periodic and the sequences of approximants in the m different arithmeticprogressions modulo m converge. The corollary below, which is also proved in [2], is an easyconsequence of Theorem 4.1. Corollary 4.6.
Let { p n } n ≥ , { q n } n ≥ be complex sequences satisfying ∞ X n =1 | p n | < ∞ , ∞ X n =1 | q n | < ∞ . ONTINUED FRACTIONS WITH MANY LIMITS 13
Let α and β be distinct roots of unity and let m be the least positive integer such that α m = β m = 1 . Define G := − αβ + q α + β + p + − αβ + q α + β + p + − αβ + q α + β + p + · · · . Let { P n /Q n } ∞ n =1 denote the sequence of approximants of G . If q n = αβ for any n ≥ , then G does not converge. However, the sequences of numerators and denominators in each ofthe m arithmetic progressions modulo m do converge. More precisely, there exist complexnumbers A , . . . , A m − and B , . . . , B m − such that, for ≤ i < m , lim k →∞ P m k + i = A i , lim k →∞ Q m k + i = B i . (23) Extend the sequences { A i } and { B i } over all integers by making them periodic modulo m sothat (23) continues to hold. Then for integers i , A i = (cid:18) A − βA α − β (cid:19) α i + (cid:18) αA − A α − β (cid:19) β i , (24) and B i = (cid:18) B − βB α − β (cid:19) α i + (cid:18) αB − B α − β (cid:19) β i . (25) Moreover, A i B j − A j B i = − ( αβ ) j +1 α i − j − β i − j α − β ∞ Y n =1 (cid:18) − q n αβ (cid:19) . (26) Put α := exp(2 πia/m ) , β := exp(2 πib/m ) , ≤ a < b < m , and r := m/ gcd( b − a, m ) .Then G has r distinct limits in b C which are given by A j /B j , ≤ j ≤ r . Finally, for k ≥ and ≤ j ≤ r , A j + kr B j + kr = A j B j . The number r occuring in this theorem is just the number of distinct limits to which thecontinued fraction tends. For this reason, we term it the rank of the continued fraction.It is easy to derive general divergence results from this theorem, including Theorem 1.1,the classical Stern-Stolz theorem [6]. The proof of Theorem 1.1 is immediate from Theorem4.1. Just set ω = 1, ω = − m = 2), q n = 0 and p n = b n . In fact, Stern-Stolzcan be seen as the beginning of an infinite family of divergence theorems. We first give ageneralization of Stern-Stolz, then give a corollary describing the infinite family. Last, welist the first few examples in the infinite family.To obtain the generalization, take q n = a n instead of q n = 0. Corollary 4.7.
Let the sequences { a n } and { b n } satisfy a n = − for n ≥ , P | a n | < ∞ and P | b n | < ∞ . Then b + K ∞ n =1 a n b n diverges. In fact, for p = 0 , , lim n →∞ P n + p = A p = ∞ , lim n →∞ Q n + p = B p = ∞ , and A B − A B = ∞ Y n =1 (1 + a n ) . Proof.
This follows immediately from Theorem 4.1, upon setting ω = 1, ω = − m = 2), q n = a n and p n = b n . (cid:3) We have not been able to find Corollary 4.7 in the literature.The natural infinite family of Stern-Stolz type theorems is described by the followingcorollary.
Corollary 4.8.
Let the sequences { a n } and { b n } satisfy a n = 1 for n ≥ , P | a n | < ∞ and P | b n | < ∞ . Let m ≥ and let ω be a primitive m -th root of unity. Then b + K ∞ n =1 − a n ω + ω − + b n does not converge, but the numerator and denominator convergents in each of the m arith-metic progressions modulo m do converge. If m is even, then for ≤ p ≤ m/ , lim n →∞ P mn + p = − lim n →∞ P mn + p + m/ = A p = ∞ , lim n →∞ Q mn + p = − lim n →∞ Q mn + p + m/ = B p = ∞ . If m is odd, then the continued fraction has rank m . If m is even, then the continued fractionhas rank m/ . Further, for ≤ p ≤ m ′ , where m ′ = m if m is odd and m/ if m is even, A p B p − − A p − B p = − ∞ Y n =1 (1 − a n ) . Proof.
In Theorem 4.1, let ω = 1 /ω . (cid:3) Some explicit examples are given below.
Example 1.
Let the sequences { a n } and { b n } satisfy a n = 1 for n ≥ , P | a n | < ∞ and P | b n | < ∞ . Then each of the following continued fractions diverges:(i) The following continued fraction has rank three: b + K ∞ n =1 − a n b n . (27) In fact, for p = 1 , , , lim n →∞ P n + p = − lim n →∞ P n + p +3 = A p = ∞ , lim n →∞ Q n + p = − lim n →∞ Q n + p +3 = B p = ∞ . (ii) The following continued fraction has rank four: b + K ∞ n =1 − a n √ b n . (28) ONTINUED FRACTIONS WITH MANY LIMITS 15
In fact, for p = 1 , , , , lim n →∞ P n + p = − lim n →∞ P n + p +4 = A p = ∞ , lim n →∞ Q n + p = − lim n →∞ Q n + p +4 = B p = ∞ . (iii) The following continued fraction has rank five: b + K ∞ n =1 − a n (1 − √ / b n . (29) In fact, for p = 1 , , , , , lim n →∞ P n + p = A p = ∞ , lim n →∞ Q n + p = B p = ∞ . (iv) The following continued fraction has rank six: b + K ∞ n =1 − a n √ b n . (30) In fact, for p = 1 , , , , , , lim n →∞ P n + p = − lim n →∞ P n + p +6 = A p = ∞ , lim n →∞ Q n + p = − lim n →∞ Q n + p +6 = B p = ∞ . In each case we have, for p in the appropriate range, that A p B p − − A p − B p = − ∞ Y n =1 (1 − a n ) . Proof.
In Corollary 4.8, set(i) ω = exp(2 πi/ ω = exp(2 πi/ ω = exp(2 πi/ ω = exp(2 πi/ (cid:3) The cases ω = exp(2 πi/m ), m = 3 , ,
10 give continued fractions that are the sameas those above after an equivalence transformation and renormalization of the sequences { a n } and { b n } . Note that the continued fractions (28) and (30) are, after an equivalencetransformation and renormalizing the sequences { a n } and { b n } , of the forms b + K ∞ n =1 − a n b n , (31)and b + K ∞ n =1 − a n b n , (32)respectively. Because of the equivalence transformations employed, the convergents do nottend to limits in (31) or (32). Also, it should be mentioned that Theorem 3.3 of [1] isessentially the special case a n = 0 of part (i) of our example. Nevertheless (31) and (32)have ranks 4 and 6 respectively.Corollary 4.6 now makes it trivial to construct q -continued fractions with arbitrarily manylimits. Example 2.
Let f ( x ) , g ( x ) ∈ Z [ q ][ x ] be polynomials with zero constant term. Let ω , ω bedistinct roots of unity and suppose m is the least positive integer such that ω m = ω m = 1 .Define G ( q ) := − ω ω + g ( q ) ω + ω + f ( q ) + − ω ω + g ( q ) ω + ω + f ( q ) + − ω ω + g ( q ) ω + ω + f ( q ) + · · · . Let | q | < . If g ( q n ) = ω ω for any n ≥ , then G ( q ) does not converge. However,the sequences of approximants of G ( q ) in each of the m arithmetic progressions modulo m converge to values in ˆ C . The continued fraction has rank m/ gcd( b − a, m ) , where a and b are as defined in Theorem 4.1. From this example we can conclude that (2) and (3) are far from unique examples andmany other q -continued fractions with multiple limits can be immediately written down.Thus, to Ramanujanize a bit, one can immediately see that the continued fractions ∞ K n ≥ − /
21 + q n and ∞ K n ≥ − / q n q n (33)both have rank four, while the continued fractions ∞ K n ≥ − /
31 + q n and ∞ K n ≥ − / q n q n (34)both have rank six.4.2. Application: Generalization of a Continued Fraction of Ramanujan.
In [3] wegave a non-trivial example of the preceding theory, the inspiration for which is a beautifulresult of Ramanujan.
Theorem 4.9.
Let | q | < , | α | = | β | = 1 , α = β , and the order of λ := α/β in T be m . For x , y = 0 and fixed | q | < , define P ( x, y ) = ∞ X n =0 x n q n ( n +1) / ( q ) n ( y q ) n . Then l.s. (cid:18) − αβα + β + q − αβα + β + q − αβα + β + q · · · (cid:19) = − βP ( qα − , βα − ) T m − αP ( qβ − , αβ − ) P ( α − , βα − ) T m − P ( β − , αβ − ) . (35) Moreover, − αβα + β + q − αβα + β + q − αβα + β + q · · · − αβα + β + q n ∼ − βP ( qα − , βα − ) λ n +1 − αP ( qβ − , αβ − ) P ( α − , βα − ) λ n +1 − P ( β − , αβ − ) . (36)In [3] we also used the Bauer-Muir Transform to produce some convergent continuedfractions. One such example is the following. ONTINUED FRACTIONS WITH MANY LIMITS 17
Corollary 1.
Let | q | < and let α and β be distinct points on the unit circle such that α/β is not a root of unity. Then − β + βqα + q + K ∞ n =2 − αβqq n + α + βq = − β ∞ X n =0 α − n q n ( n +3) / ( q ; q ) n ( βq/α ; q ) n ∞ X n =0 α − n q n ( n +1) / ( q ; q ) n ( βq/α ; q ) n . (37)5. Poincar´e type recurrences
Let the sequence { x n } n ≥ have the initial values x , . . . , x p − and be subsequently definedby x n + p = p − X r =0 a n,r x n + r , (38)for n ≥
0. Suppose also that there are numbers a , . . . , a p − such thatlim n →∞ a n,r = a r , ≤ r ≤ p − . (39)A recurrence of the form (38) satisfying the condition (39) is called a Poincar´e-type recur-rence, (39) being known as the Poincar´e condition. Such recurrences were initially studiedby Poincar´e who proved that if the roots of the characteristic equation t p − a p − t p − − a p − t p − − · · · − a = 0 (40)have distinct norms, then the ratios of consecutive terms in the recurrence (for any set ofinitial conditions) tend to one of the roots. See [8]. Because the roots are also the eigenvaluesof the associated companion matrix, they are also referred to as the eigenvalues of (38). Thisresult was improved by O. Perron, who obtained a number of theorems about the limitingasymptotics of such recurrence sequences. Perron [7] made a significant advance in 1921when he proved the following theorem which for the first time treated cases of eigenvalueswhich repeat or are of equal norm. Theorem 5.1.
Let the sequence { x n } n ≥ be defined by initial values x , . . . , x p − and by (38)for n ≥ . Suppose also that there are numbers a , . . . , a p − satisfying (39). Let q , q , . . . q σ be the distinct moduli of the roots of the characteristic equation (40) and let l λ be the numberof roots whose modulus is q λ , multiple roots counted according to multiplicity, so that l + l + . . . l σ = p. Then, provided a n, be different from zero for n ≥ , the difference equation (38) has afundamental system of solutions, which fall into σ classes, such that, for the solutions of the λ -th class and their linear combinations, lim sup n →∞ n p | x n | = q λ . The number of solutions of the λ -th class is l λ . Thus when all of the characteristic roots have norm 1, this theorem gives thatlim sup n →∞ n p | x n | = 1 . Another related paper is [4] where the authors study products of matrices and give asufficient condition for their boundedness. This is then used to study “equimodular” limitperiodic continued fractions, which are limit periodic continued fractions in which the charac-teristic roots of the associated 2 × Theorem 5.2.
Let the sequence { x n } n ≥ be defined by initial values x , . . . , x p − and by x n + p = p − X r =0 a n,r x n + r , (41) for n ≥ . Suppose also that there are numbers a , . . . , a p − such that ∞ X n =0 | a r − a n,r | < ∞ , ≤ r ≤ p − . Suppose further that the roots of the characteristic equation t p − a p − t p − − a p − t p − − · · · − a = 0 (42) are distinct and all on the unit circle, with values, say, α , . . . , α p − . Then there existcomplex numbers c , . . . , c p − such that x n ∼ p − X i =0 c i α ni . (43)The following corollary, also proved in [2], is immediate. Corollary 5.3.
Let the sequence { x n } n ≥ be defined by initial values x , . . . , x p − and by x n + p = p − X r =0 a n,r x n + r , for n ≥ . Suppose also that there are numbers a , . . . , a p − such that ∞ X n =0 | a r − a n,r | < ∞ , ≤ r ≤ p − . Suppose further that the roots of the characteristic equation t p − a p − t p − − a p − t p − − · · · − a = 0 ONTINUED FRACTIONS WITH MANY LIMITS 19 are distinct roots of unity, say α , . . . , α p − . Let m be the least positive integer such that,for all j ∈ { , , . . . , p − } , α mj = 1 . Then, for ≤ j ≤ m − , the subsequence { x mn + j } ∞ n =0 converges. Set l j = lim n →∞ x nm + j , for integers j ≥ . Then the (periodic) sequence { l j } satisfies the recurrence relation l n + p = p − X r =0 a r l n + r , and thus there exist constants c , · · · , c p − such that l n = p − X i =0 c i α ni . Applications to ( r, s ) -matrix continued fractions In [5], the authors define a generalization of continued fractions called ( r, s )-matrix contin-ued fractions. This generalization unifies a number of generalizations of continued fractionsincluding “generalized (vector valued) continued fractions” and “G-continued fractions”, see[6] for terminology.Here we show that our results apply to limit periodic ( r, s )-matrix continued fractions witheigenvalues of equal magnitude, giving estimates for the asymptotics of their approximantsso that their limit sets can be determined.For consistency we closely follow the notation used in [5] to define ( r, s )-matrix continuedfractions. Let M s,r ( C ) denote the set of s × r matrices over the complex numbers. Let θ k be a sequence of n × n matrices over C . Assume that r + s = n . A ( r, s )-matrix continuedfraction is associated with a recurrence system of the form Y k = Y k − θ k . The continuedfraction is defined by its sequence of approximants. These are sequences of s × r matricesdefined in the following manner.Define the function f : D ∈ M n ( C ) → M s,r ( C ) by f ( D ) = B − A, (44)where B is the s × s submatrix consisting of the last s elements from both the rows andcolumns of D , and A is the s × r submatrix consisting of the first r elements from the last s rows of D .Then the k -th approximant of the ( r, s )-matrix continued fraction associated with thesequence θ k is defined to be s k := f ( θ k θ k − · · · θ θ ) . (45)To apply Theorem 1 to this situation, we endow M s × r ( C ) with a metric by letting thedistance function for two such matrices be the maximum absolute value of the respectivedifferences of corresponding pairs of elements. Then, providing that the f is continuous,(a suitable specialization of) our theorem can be applied. (Note that f will be continuousproviding that it exists, since the inverse function of a matrix is continuous when it exists.)Let lim k →∞ θ k = θ , for some θ ∈ M n ( C ). Then the recurrence system is said to be ofPoincar´e type and the ( r, s )-matrix continued fraction is said to be limit periodic. Underour usual condition, Theorem 1 can be applied and the following theorem results. Theorem 6.1.
Suppose that the condition P k ≥ || θ k − θ || < ∞ holds, that the matrix θ is di-agonalizable, and that the eigenvalues of θ are all of magnitude . Then the k th approximant s k has the asymptotic formula s k ∼ f ( θ k F ) , (46) where F is the matrix defined by the convergent product F := lim k →∞ θ − k θ k θ k − · · · θ θ . Note that because of the way that ( r, s )-matrix continued fractions are defined, we havetaken products in the reverse order than the rest of the paper.As a consequence of this asymptotic, the limit set can be determined from l.s. ( s k ) = l.s. ( f ( θ k F )) . Conclusion
Because of length restrictions, we have omitted several corollaries as well as most proofs.Interested readers should consult the author’s papers [2] and [3].
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Northern Illinois University, Mathematical Sciences, DeKalb, IL 60115-2888
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