On a continued fraction formula of Wall

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1 c Te Ramanujan Journal,, 7 () Kluwer Academic Publisers, Boston. Manufactured in Te Neterlands. On a continued fraction formula of Wall DONGSU KIM * Department of Matematics, KAIST, Taejon , Korea dskim@mat.kaist.ac.kr JIANG ZENG zeng@desargues.univ-lyon.fr Institut Girard Desargues, Université Claude Bernard (Lyon I), Villeurbanne, France Editor: Mourad Ismail Abstract. We study te combinatorics of a continued fraction formula due to Wall. We also derive te ortogonality of little q-jacobi polynomials from tis formula, as Wall did for little q-laguerre polynomials. Keywords: Dyck pat, continued fraction formula, little q-jacobi polynomials. Introduction Wall [7] found te following continued fraction formula: ( g )t ( g 2 )g t ( g 3)g 2 t ( g 4)g 3 t... g t ( g )g 2 t ( g 2)g 3 t ( g 3)g 4 t... = t. () From te modern combinatorial point of view [3, 6], te above identity can be seen as an equation between generating functions of certain weigted Dyck pats. So it is natural to seek for a bijective proof of te identity in tis context. We give suc a proof in sections 2 and 3. One of te consequences of () is te discovery of te little q-laguerre polynomials, wic were te precursor of te little q-jacobi polynomials studied by Andrews and Askey []. Recall tat te monic little q-jacobi polynomials [4, p.92] are defined by te recurrence relation: xp n (x) = p n+ (x) + (A n + C n )p n (x) + A n C n p n (x), n, were p (x) = 0, p 0 (x) = and A n = qn ( aq n+ )( abq n+ ) ( abq 2n+ )( abq 2n+2 ), C n = aqn ( q n )( bq n ) ( abq 2n )( abq 2n+ ). (2) * Tis work was partially supported by KOSEF: and

2 2 D. KIM AND J. ZENG + + (0, 0) (8, 0) Figure. Up and down steps, and a Dyck pat of lengt 8. Te explicit formula of p n (x) is p n (x) = (aq; q) n (abq n+ ; q) n ( ) n q (n 2) 2 φ (q n, abq n+ ; aq q; qx). As sown recently by Ismail and Stanton [5] te little q-jacobi polynomials are ortogonal wit respect to te moment sequence {µ n } n 0, were µ n = (aq; q) n (abq 2 ; q) n. In view of (2), tis ortogonality is equivalent to te following continued fraction expansion: µ n t n =, (3) λ t n 0 λ 2 t λ 3t... were λ 2n = A n and λ 2n = C n for n. In section 4 we prove tat (3), wic is equivalent to te ortogonality of te little q-jacobi polynomials, follows from (), as Wall [8] did for te special b = 0 case. 2. Combinatorial model A Dyck pat [3, 6] of lengt 2n can be described as a sequence (y 0, y,..., y 2n ) of non-negative integers suc tat y 0 = y 2n = 0, and y i = y i ±, i 2n. It can also be caracterized by its step sequence s = (s, s 2,..., s 2n ), were s i = u, if y i = y i +, and s i = d, if y i = y i, i 2n. Call s i an up step if s i = u and a down step if s i = d. Te level of te step s i is defined to be y i. A Dyck pat can be visualized as a connected pat in te plane R 2 from (0, 0) to (2n, 0), by drawing a line segment between consecutive points (i, y i ), 0 i 2n. (See Figure.) To combinatorially interpret () we need te notion of weigted Dyck pats. Tere are two types of weigt functions w () and w (2) on te steps s i, starting from level : { w () g, if s (s i ) = i is up, or g, if s i is down,

3 WALL S CONTINUED FRACTION FORMULA 3 g g + g g Type Type 2 Figure 2. 0, g 0 = in type, and g 0 = in type 2. were g 0 =, and w (2) (s i ) = { g+, if s i is up, or g, if s i is down, were g 0 =. (See Figure 2.) A weigted Dyck pat is ten defined as a pair P = (s, w), were s is a sequence of steps forming a Dyck pat and w is a sequence of weigts of type or 2 of te entries in s. Te sequence s is called te step sequence of P and te sequence w te weigt sequence of P. Te weigt of P = (s, w) is te product of entries in w. Let P n (i) be te set of weigted Dyck pats of lengt 2n wit weigt of type i, i = or 2. Set P n = n k=0 ( ) P () k P (2) n k, were P () 0 = P (2) 0 = {(, )}, te set of te empty weigted Dyck pat of lengt 0 wose weigt is. We now put a weigt w on P n by w(x) = w () (P )w (2) (Q), if x = (P, Q) P n. Ten formula () is equivalent to te assertion w(p n ) = x P n w(x) =, for n 0. (4) We will construct a weigt-preserving sign-reversing (wpsr) involution φ on te set P n, i.e. φ is an involution and w(φ(x)) = w(x), if φ(x) x, wit one fixed point of weigt, wic proves (4) since w(x) = w(x) =. (5) x P n x: φ(x)=x 3. Involution for Wall s formula For clarity we first define two simple involutions φ and φ 2 and build te involution φ as teir composition. Let F n denote te set of pats in P n () suc tat tere is no step wit weigt ±g ( ) followed by steps wit weigt. We define an involution φ on P n () wit fixed set F n, i.e. te set of fixed points of φ. Let P = (s, w) be a weigted pat in P n () \ F n. Let k be te smallest integer suc tat w k = ±g for some ( ) and w k+ = w k+2 = = w k+ =. Let φ (P ) be te weigted pat in P n () \ F n

4 4 D. KIM AND J. ZENG g 3 g 3 k k + + k k + + φ Figure 3. Involution φ on P () n. wit weigt sequence w = (w,..., w k, g, w k+,..., w 2n ). Te step sequence of φ (P ) is determined by te new weigt sequence w. Te map φ is illustrated in Figure 3. Clearly we ave te following result. Lemma Te mapping φ is a wpsr involution on P () n wose fixed set is F n. Let P = (s, w) be a weigted Dyck pat in F n. If s 2i+ = u is te rigtmost up step starting from level 0, ten s 2j = u, s 2j = d and w 2j = w 2j = for j i. Te pat P is called reduced, if i = 0. In particular, P re = ((s 2i+, s 2i+2,..., s 2n ), (w 2i+, w 2i+2,..., w 2n )) is a reduced pat in F n i, called te reduced component of P. Given a reduced pat P = (s, w) in F n, let (i, j ), (i 2, j 2 ),..., (i l, j l ) be all te pairs of integers suc tat (w ik, w ik +,..., w jk ) = (g k,,..., ), for k > 0, and w jk +. We call te segment (w ik, w ik +,..., w jk ) an up-down ook, since an up step is followed by down steps. Note tat i < j < i 2 < j 2 < < i l < j l. We define a weigted pat P = ((s, s 2,..., s 2n), (w, w2,..., w2n)) in P n (2), called te sadow of P, as follows: d, if i {i, i 2,..., i k }, s i = u, if i {j, j 2,..., j k }, s i, oterwise, { wi+, if s wi i = s i+, = w i+, if s i s i+,, if i = 2n. Note tat P F n is reduced if and only if P doesn t touc te level 0 except at te two end points, and tat if w i+ = ten wi =. A Dyck pat is said to be prime if it doesn t touc te level 0 except for te two end points and a weigted pat is said to be prime if its underlying pat is prime. Given a prime weigted pat Q = (s, w) in P n (2), let (i, j ), (i 2, j 2 ),..., (i l, j l ) be all te pairs suc tat (w ik, w ik +,..., w jk ) = (,...,, g k ), for k > 0, and w ik. We call te segment (w ik, w ik +,..., w jk ) a down-up ook, since a down step is followed by up steps. Note again tat i < j < i 2 < j 2 < < i l < j l. We define a weigted pat Q = ((s, s 2,..., s 2n), (w, w2,..., w2n)) in F n, called te

5 WALL S CONTINUED FRACTION FORMULA 5 g 3 g g 3 4 g 2 g 2 g P : 2 g 3 g g g 2 g g 4 g 3 Q : g 3 g 3 g 2 g 2 g 2 g g 2 g g Figure 4. P = Q and Q = P. sade of Q, as follows: u, if i {i, i 2,..., i k }, s i = d, if i {j, j 2,..., j k }, s i, oterwise,, if i =, wi = w i, w i, if s i = s i, if s i s i. Sadow and sade are illustrated in Figure 4. Te line segments wic are different from tose in te original pat are represented as dotted lines. Note tat eac up-down ook in P becomes a down-up ook in P and eac down-up ook in Q becomes a up-down ook in Q. Lemma 2 Te sadow and sade are well-defined and te mapping : P P is an injection from te set of reduced pats in F n to P n (2) suc tat w (2) (P ) = w () (P ). Proof: We need to sow tat P and Q are legitimate pats in P n (2) and F n, respectively. Comparing te weigts of type and type 2, it is easy to see tat if s i = s i or i {i, i 2,..., i k }, ten te step s i can be weigted by wi in type 2. If i {j, j 2,..., j k }, ten s i = u and w i, wic guarantee tat s i can be weigted by wi. Hence P P n (2) is well-defined. Tis is illustrated in Figure 4. Te case of Q is similar and so we omit. Since w is a rearrangement of te entries in w wit some canges in sign, we first note tat w (2) (P ) = w () (P ). So it suffices to sow tat te signs of w (2) (P ), w () (P ) differ, wic follows from te fact tat te number of down steps wit weigt in P is one less tan te number of tose in P, because w = is an up step in P wile w2n = w is a down step in P.

6 6 D. KIM AND J. ZENG g 2 g g 2 3 P g g 2, g 2 Q g g g φ g 3 g 2 P g 2 g 2 g 2 Q g g g g, Figure 5. φ 2 : (P, Q) (P, Q ) It remains to sow tat te correspondence is injective. Let Q be a pat in P (2) n suc tat Q = P for some reduced pat P in F n. Ten we can sow tat Q = P, wic implies tat is one-to-one. Details are omitted. Let X n = n k=0 (F k P (2) n k ). We define te involution φ 2 on X n as follows. For any pair (P, Q) in X n, if te reduced component P re of P is not empty, ten cut off P re from P and attac P re to Q from te left; if P re is empty but Q is not empty, ten cut off te leftmost prime component Q pr of Q, wic is te initial segment of Q tat comes back to te level 0 for te first time, and attac Q pr to P from te rigt; if bot P re and Q are empty, i.e. (P, Q) = (((u, d;... ; u, d), (, ;... ;, )), (, )), ten do noting. Let φ 2 (P, Q) be te resulting pair and extend te domain of φ 2 to P n by setting φ 2 (P, Q) = (P, Q) for (P, Q) P n \ X n. Ten φ 2 is a wpsr involution on P n, wic is illustrated in Figure 5. Finally, combining te involutions φ and φ 2, we define te mapping φ on P n by φ(p, Q) = φ 2 (φ (P ), Q) and ave te following result. Teorem For any integer n 0, te mapping φ is a wpsr involution on P n, wic as te unique fixed point (((u, d;... ; u, d), (, ;... ;, )), (, )) of weigt. Obviously (4) and ten Wall s formula () follow from te above teorem. 4. Ortogonality of little q-jacobi polynomials Note first tat te coefficients λ n in (3) can be parameterized as follows : were g 0 = and for n λ n = ( g n )g n, n, g 2n = aqn ( bq n ) abq 2n, g 2n = qn ( abq n+ ) abq 2n+.

7 WALL S CONTINUED FRACTION FORMULA 7 Consider te formal power series φ(b, t) = a n (b)t n = ( g )t n 0 ( g 2)g t ( g 3)g 2 t Wall s formula () implies ten.... (6) ( t)φ(b, t) = g t ( g )g 2 t ( g 2)g 3 t... = aq( bq)t φ(bq, qt). abq2 Equating te coefficients of t n yields tat a 0 (b) = and a n (b) = a n (b) aqn ( bq) abq 2 a n (bq). (7) From tis identity we derive readily tat a n (b) = (aq; q) n (abq 2 ; q) n = µ n. Since te continued fractions in (3) and (6) are identical, tis proves (3). References. Andrews (G.) and Askey (R.), Enumeration of partitions: te role of Eulerian series and q-ortogonal polynomials. In : Higer Combinatorics (ed. M. Aigner), Reidel Publications, Boston, 977, Ciara (T. S.), An introduction to ortogonal polynomials, Matematics and its Applications, Vol. 3. Gordon and Breac Science Publisers, New York-London-Paris, Flajolet (P.), Combinatorial aspects of continued fractions, Disc. Mat., 32 (980), Koekoek (R.) and Swarttouw (R. F.), Te Askey-sceme of ypergeometric ortogonal polynomials and its q-analogue, Delft University of Tecnology, Report Ismail (M. E. H.) and Stanton (D.), More ortogonal polynomials as moments, In: Matematical essays in onor of Gian-Carlo Rota (Cambridge, MA, 996), , Progr. Mat., 6, Birkäuser Boston, MA, Toucard (J.), Sur un problème de configurations et sur les fractions continues, Canadian J. Mat. 4, (952) Wall (H.S.), Continued fractions and totally monotone sequences, Trans. Amer. Mat. Soc., 48 (940), Wall (H.S.), A continued fraction related to some partition formulas of Euler, Amer. Mat. Montly, 59 (952),