Properties of matrix orthogonal polynomials via their Riemann-Hilbert characterization

Symme y, In eg abili y and Geome y: Me hods and Applica ions SIGMA 7(2011), 098, 31 pages P ope ies o Ma ix O hogonal Polynomials ia hei Riemann–Hilbe Cha ac e iza ion F. Albe o GR ¨ UNBAUM †, Manuel D. DE LA IGLESIA ‡ and And ei MART´ INEZ-FINKELSHTEIN § †Depa men o Ma hema ics, Uni e si y o Cali o nia, Be keley, Be keley, CA 94720 USA E-mail: g unb[email p o ec ed]e keley.edu URL: h p://ma h.be keley.edu/~g unbaum/ ‡Depa amen o de An´alisis Ma em´a ico, Uni e sidad de Se illa, Apdo (P. O. BOX) 1160, 41080 Se illa, Spain E-mail: [email p o ec ed] URL: h p://eule .us.es/~mdi29/ §Depa amen o de Es ad´ıs ica y Ma em´a ica Aplicada, Uni e sidad de Alme ´ıa, 04120 Alme ´ıa, Spain E-mail: and [email p o ec ed] URL: h p://www.ual.es/~and ei/ Recei ed June 09, 2011, in inal o m Oc obe 20, 2011; Published online Oc obe 25, 2011 h p://dx.doi.o g/10.3842/SIGMA.2011.098 Abs ac . We gi e a Riemann–Hilbe app oach o he heo y o ma ix o hogonal poly- nomials. We will ocus on he algeb aic aspec s o he p oblem, ob aining di e ence and di e en ial ela ions sa is ied by he co esponding o hogonal polynomials. We will show ha in he ma ix case he e is some ex a eedom ha allows us o ob ain a amily o lad- de ope a o s, some o hem o 0- h o de , some hing ha is no possible in he scala case. The combina ion o he ladde ope a o s will lead o a amily o second-o de di e en ial equa ions sa is ied by he o hogonal polynomials, some o hem o 0- h and i s o de , some hing also impossible in he scala se ing. This shows ha he di e en ial p ope ies in he ma ix case a e much mo e complica ed han in he scala si ua ion. We will s udy se e al examples gi en in he las yea s as well as o he s no conside ed so a . Key wo ds: ma ix o hogonal polynomials; Riemann–Hilbe p oblems 2010 Ma hema ics Subjec Classi ica ion: 42C05; 35Q15 1 In oduc ion The heo y o ma ix o hogonal polynomials on he eal line (MOPRL) has i s ounda ions in he seminal pape s o K ein [42,43] (see also hei accoun in he book o Be ezans’ki˘ı [2]). Fo u he his o ical backg ound and analy ic esul s, he eade is e e ed o he su ey [11] and o Chap e 4 o [51]. In many aspec s he MOPRL esemble hei scala coun e pa s, especially whe e he p oo s a e based on basic p ope ies o Hilbe spaces. Ne e heless, he non-commu a i i y o ma ix mul iplica ion and he exis ence o non-ze o singula ma ices add ea u es o he heo y ha make MOPRL an in e es ing objec o s udy. Mo eo e , many p ob- lems o scala polynomials a e be e unde s ood o ecas in e ms o some ma ix polynomials, see, o ins ance, [26]. Recall ha a ma ix polynomial o deg ee ≤nin CN×Nand a scala a iable xcan be de ined as an exp ession o he o m Anxn+· · · +A1x+A0, 2 F.A. G ¨unbaum, M.D. de la Iglesia and A. Ma ´ınez-Finkelsh ein whe e Aj’s a e cons an ma ices in CN×N. In wha ollows we conside N ixed, and deno e by Pn he amily o ma ix polynomials o deg ee ≤nin CN×N, as well as by P:= Sn≥0Pn. We use p e e ably bold ace le e s o deno e ma ices, and s anda d on o scala s. We also use IN o he N×Niden i y ma ix, omi ing he explici e e ence o i s dimension when i canno lead anyone in o con usion. In his pape we conside some aspec s o he MOPRL heo y, assuming ha he o hogonali y is gi en by an absolu ely con inuous measu e on he line. Mo e p ecisely, ou s a ing poin is a weigh W= (Wij) : (a, b)→GL(N, R), de ined and posi i e de ini e on a ini e o in ini e in e al (a, b)⊂R. We will assume ha all Wij and W0 ij ha e ini e momen s: Zb a |x|nW(x)dx < ∞,Zb a |x|nW0(x)dx < ∞, n ∈N0:= N∪ {0}, whe e he in eg a ion o a ma ix unc ion is applied en y-wise. Fo any wo P,Q∈P, he weigh Winduces wo ma ix- alued “inne p oduc s”, (P,Q)W=Zb a P(x)W(x)Q∗(x)dx, and hP,QiW=Zb a Q∗(x)W(x)P(x)dx = (Q∗,P∗)W, whe e he as e isk deno es he conjuga e anspose (o He mi ian conjuga e) o a ma ix. Due o his connec ion be ween bo h inne p oduc s, we es ic ou a en ion o (·,·)W. We de ine also he no m kPkW:= (T hP,PiW)1/2, and assume ha Wis non- i ial, in he sense ha kPkW>0 o e e y non-ze o ma ix polynomial P. In his case (see [11, Lemma 2.3] o [51, P oposi ion 4.2.3]), (P,P)Wis non- singula o e e y non-ze o polynomial P, and we can easily implemen a ma ix analogue o he G am–Schmid o hogonaliza ion p ocedu e, which yields a unique sequence ( b Pn)no monic o hogonal polynomials such ha b P0=I, b Pn(x) = xnI+ n−1 X j=0 an,jxj,(b Pn,Q)W=0 o e e y Q∈Pn−1, n ∈N,(1.1) as well as ma ix polynomials (Pn)n, “o hono mal” wi h espec o W, such ha Pn(x) = κnb Pn,(Pn,Pm)W=δn,mI.(1.2) Ob iously, Pn’s a e de e mined up o a uni a y le ac o , so we can speak abou an equi alence class o o hono mal MOPRL, co esponding o he weigh W. Addi ionally, i Bis a cons an non-singula ma ix, hen W(x) = BW (x)B∗ is also a weigh , and ( e Pn)n= (PnB−1)nis he co esponding sequence o o hono mal polyno- mials. Hence, mo e han a single weigh we conside an equi alence class o weigh s gi en by i s ep esen a i e W. In pa icula , wi hou loss o gene ali y we can assume ha W(x0) = I a a poin x0∈(a, b). I his class con ains a diagonal ma ix- alued unc ion, we say ha W P ope ies o Ma ix O hogonal Polynomials ia hei Riemann–Hilbe Cha ac e iza ion 3 educes o scala weigh s. A cha ac e iza ion o his ac , as shown in [23], is he commu a i i y condi ion W(x)W(y) = W(y)W(x), x, y ∈[a, b]. This si ua ion is conside ed i ial, and usually omi ed om conside a ion. Wo k in he las ew yea s has e ealed a numbe o explici amilies o MOPRL; in many cases hey a e join eigen unc ions o some ixed di e en ial ope a o wi h ma ix coe icien s independen o he deg ee no he MOPRL. This s udy was ini ia ed in [21], bu non i ial examples had o wai un il [23,29,34,36]. A solu ion o he classi ica ion p oblem o hose W whose MOPRL a e common eigen unc ions o some ixed di e en ial ope a o emains elusi e. The e a e by now wo me hods ha ha e yielded non i ial examples: a) he connec ion wi h ma ix- alued sphe ical unc ions o symme ic spaces [34,36], and b) a combina ion o classical me hods and some Lie algeb a ools [23,24]. A necessa y condi ion in e ms o momen s is gi en in [23], and a necessa y and su icien condi ion in e ms o he “ad-condi ions” is gi en in [38]. In his app oach one ge s an explici o mula o one di e en ial ope a o . Conce ning applica ions, he ecu en ela ions ha e been used in he s udy o ce ain quasi-bi h-and-dea h p ocesses [30,31,32,33], in which case he di e en ial ope a o plays no ole. Recen ly, new applica ions o hese p ocesses ha e been ela ed o u n o Young diag ams models [37]. A big boos in he esea ch o he scala OPRL has been hei Riemann–Hilbe (RH) cha- ac e iza ion, in oduced in he seminal pape o Fokas, I s and Ki ae [28], and complemen ed wi h a non-linea s eepes descen analysis in a se ies o wo ks o Dei , Zhou and collabo a- o s [12,13,16,17]. This combina ion has allowed o es ablish ex emely s ong asymp o ic esul s wi h applica ions in andom ma ix heo y, in pa icula o uni a y in a ian ensem- bles [12,14], de e minan al poin p ocesses [8,9,44,45], o hogonal Lau en polynomials [46,47], Painle ´e ascenden s [10,41], he Toda la ices [15], o men ion a ew. The RH cha ac e iza ion, e en wi hou he s eepes descen analysis, allows one o p o e o he algeb aic and analy ic p ope ies o o hogonal polynomials, as i was illus a ed in [13] o OP on he uni ci cle, and in [40, Chap e 22], o classical amilies o OPRL. One o he goals o his pape is o ex end hese conside a ions o MOPRL, some hing ha , su p isingly enough, has no been explo ed in dep h so a . We will show ha he RH o mula ion, which has a e y na u al block-wise gene aliza ion o he MOPRL case, allows one o e eal some sub le ies hidden in he ma ix case. Fo ins ance, we will de i e iden i ies and di e ence-di e en ial equa ions ha we e unknown e en o some o he explici amilies men ioned abo e. The RH echnique can be also a na u al me hod o ob aining he di e en ial ela ions o MOPRL wi h espec o gene al weigh ma ices. The e ha e been se e al pape s conside ing he block-wise RH p oblem o he la o simila o he one discussed in his pape , bu o singula (namely, ank 1) weigh ma ices. Fo hese weigh s he MOPRL a e connec ed wi h a class o mul iple o hogonal polynomials ha ind applica ions in he analysis o de e minan al poin p ocesses and non-in e sec ing s ochas ic pa hs (see e.g. he wo ks o Kuijlaa s and collabo a o s [8,18,19,45]) o o he mul icomponen 2D Toda la ice hie a chy [1]. This pape deals wi h s ic ly algeb aic consequences o he RH o mula ion o ma ix poly- nomials o hogonal wi h espec o a weigh ma ix suppo ed on an in e al [a, b]⊂R; he asymp o ic analysis ia he Dei –Zhou non-linea s eepes descen me hod will no be discussed he e. In Sec ion 2we will discuss wo dual RHPs, which a e uniquely sol ed in e ms o he MOPRL. S anda d a gumen s ela ed o RHP allow us o ob ain he h ee- e m ecu ence ela ion and he di e en ial iden i ies (ladde ope a o s) o MOPRL. Bo h yield also he so-called Lax pai o he polynomials, which is an o e -de e mined sys em (2.28), whose compa ibili y condi ions 4 F.A. G ¨unbaum, M.D. de la Iglesia and A. Ma ´ınez-Finkelsh ein ende u he p ope ies o MOPRL. All he esul s in his sec ion ha e been p e iously ob ained using di e en app oaches, bu he RH me hod will p o ide new and s aigh o wa d p oo s. In Sec ion 3we will conside a special ans o ma ion o he o iginal RHP, based on ac o i- za ions o he weigh ma ix o he o m W=T T ∗. The main goal is o ob ain a RHP wi h cons an jumps (independen o z), in o de o simpli y he di e en ial ela ion ob ained ea lie . Fo simplici y, we will ocus on he case whe e he suppo o he weigh ma ix is R. The ex ension o he me hod o weigh s suppo ed on semi-in ini e o ini e in e als is simple, see a sho discussion a he end o his pape . We exploi also he non-uniqueness o he weigh ac o iza ion, which in he ma ix case gi es us some ex a eedom and yields non- i ial ela ions. In pa icula , a amily o ladde ope a o s, some o hem o 0- h o de , is ob ained, a phenomenon ha is no possible in he scala si ua ion. Fu he mo e, by combining app op ia ely he amily o ladde ope a o s, we will ge he 0- h, i s and second o de di e en ial equa ions sa is ied by he MOPRL. Some examples o lowe o de di e en ial ope a o s appea in [5,6,35]. The conside a ions so a ha e been comple ely gene al, applicable o any amily o MOPRL. In Sec ion 4we na ow he analysis o Sec ions 2and 3 o some ele an examples o MOPRL suppo ed on R, ob aining new esul s e en in he cases s udied p e iously in he li e a u e. In he inal s age o p epa a ion o his manusc ip we lea ned abou he p ep in [4] which also uses he Riemann–Hilbe app oach o he analysis o he ma ix o hogonal polynomial. Al hough he e is some o e lapping be ween he esul s con ained in ou Sec ions 2,3and in [4, Sec ion 2], he ocus o bo h con ibu ions is di e en , and in his sense, complemen a y. 2 The Riemann–Hilbe p oblem o MOPRL 2.1 Fo mula ion and basic p ope ies In his sec ion we discuss he Riemann–Hilbe p oblem (RHP) ela ed o MOPRL wi h espec o a N×Nweigh ma ix Wsuppo ed on an in e al [a , b ]⊂R; his in e al can be ei he bounded o unbounded. We assume Wcon inuous and non- anishing on (a, b), and ha a any ini e endpoin o he suppo he weigh Whas a wo se a powe - ype singula i y, ha is, W is o he o m W(z) = |z−c|γc W(z), γc>−1,(2.1) whe e c∈ {a, b},c6=±∞, and Wis a bounded, con inuous and non- anishing a z=c. This class o weigh s comp ises all he examples conside ed so a in he li e a u e. As a con en ion, in wha ollows we w i e he 2N×2Nma ices pa i ioned in o N×N blocks, as in (2.2) below. Recall ha A∗s ands o he He mi ian conjuga e o he ma ix A, as well as A−∗ = (A∗)−1. We adop he con en ion ha o any ma ix- alued unc ion P(z) o a complex a iable z,P∗deno es he ma ix- alued unc ion ob ained as P∗(z) := (P(¯z))∗. The RHP o MOPRL wi h espec o a weigh ma ix Wconsis s in inding a ma ix unc ion Yn:C→C2N×2Nsuch ha (Y1) Ynis analy ic in C [a, b]. (Y2) Ynhas on (a, b) con inuous bounda y alues Yn +( esp., Yn −) om he uppe ( esp., lowe ) hal plane, such ha Yn +(x) = Yn −(x)INW(x) 0IN, x ∈(a, b).(2.2) P ope ies o Ma ix O hogonal Polynomials ia hei Riemann–Hilbe Cha ac e iza ion 5 (Y3) As z→ ∞, o e e y m∈Nwe ha e Yn(z) = I2N+ m X i=1 Yn i zi+O1/zm+1! znIN0 0z−nIN!(2.3) (whe e he asymp o ic e m O(1/zm+1) depends on n). (Y4) As z→c,c∈ {a, b},c6=±∞, we ha e Yn(z) = O(1) O(h(z)) O(1) O(h(z)) !, whe e h(z) =      |z−c|γc,i −1< γc<0, log |z−c|,i γc= 0, 1,i γc>0. (2.4) Rema k 2.1. Usually he asymp o ic condi ion a in ini y (2.3) is s a ed o m= 1; howe e , i can be p o ed ha (2.3) o m= 1 implies ha his condi ion holds o e e y m∈N. Along wi h he RH p oblem (Y1)–(Y4) we can conside he ollowing dual p oblem: inding a ma ix unc ion yn:C→C2N×2Nsuch ha (y1) ynis analy ic in C [a, b]. (y2) yn +(x) = IN−W(x) 0INyn −(x) when x∈(a, b). (y3) As z→ ∞, o e e y m∈Nwe ha e yn(z) = z−nIN0 0znIN! I2N+ m X i=1 e Yn i zi+O1/zm+1!.(2.5) (y4) As z→c,c∈ {a, b},c6=±∞, we ha e yn(z) = O(h(z)) O(h(z)) O(1) O(1) !, wi h hde ined in (2.4). I will u n ou ha (y1)–(y4) is ela ed o he in e se o he solu ion o (Y1)–(Y4). Fo any in eg able N×Nma ix- alued unc ion Fon [a, b], C(F)(z) := 1 2πi Zb a F( ) −zd de ines he Cauchy o S iel jes ans o m o F, which is a ma ix- alued and analy ic unc ion in C [a, b]. Le us in oduce he ma ix polynomials o he second kind (o deg ee n−1), de ined by Qn(x) = Zb a Pn( )−Pn(x) −xW( )d , n ≥0.(2.6) 6 F.A. G ¨unbaum, M.D. de la Iglesia and A. Ma ´ınez-Finkelsh ein We ha e 2πiC(PnW)(z) = Qn(z)+2πiPn(z)C(W)(z), whe e 2πiC(W) is he m- unc ion o he weigh ma ix W. Finally, i κnis he leading coe icien o any co esponding no malized polynomial Pn, we deno e γn=κ∗ nκn.(2.7) Obse e ha a p io i γndepends on he selec ion o κn. The ollowing is a comple e analogue o he well-known heo em om [28], al hough i s p oo equi es some addi ional conside a ions: Theo em 2.2. The unique solu ion o he RHP (Y1)–(Y4) is Yn(z) = Rn(z)Y0(z), n ≥0,(2.8) whe e Y0(z) = INC(W)(z) 0IN, and he ans e ma ix Rnis a ma ix polynomial gi en by R0(z) = I, Rn(z) = κ−1 nPn(z) (2πiκn)−1Qn(z) −2πiκ∗ n−1Pn−1(z)−κ∗ n−1Qn−1(z)!, n ∈N.(2.9) Analogously, he unique solu ion o he RHP (y1)–(y4) is yn(z) = y0(z) n(z), n ≥0,(2.10) whe e y0(z) = IN−C(W)(z) 0IN, and he ans e ma ix nis a ma ix polynomial gi en by 0(z) = I, n(z) = −Q∗ n−1(z)κn−1−(2πi)−1Q∗ n(z)κ−∗ n 2πiP∗ n−1(z)κn−1P∗ n(z)κ−∗ n!, n ∈N.(2.11) Mo eo e , o all n≥0, yn(z) = 0−IN IN0(Yn(z))T0IN −IN0= (Yn(z))−1,(2.12) de Yn(z)=1, o all z∈C.(2.13) He e ATdeno es he anspose o he ma ix A. Rema k 2.3. In he scala case, when Ynis a 2 ×2 ma ix, he e is no need o conside simul- aneously bo h RHPs (Y1)–(Y4) and (y1)–(y4) since he exis ence o (Yn)−1and (2.11), (2.12) ollow di ec ly om (2.13). In he gene al (2N)×(2N) case mo e ca e should be pu in he analysis o he local beha io a he endpoin s a,b, and (2.12) is no longe s aigh o wa d. P ope ies o Ma ix O hogonal Polynomials ia hei Riemann–Hilbe Cha ac e iza ion 7 Rema k 2.4. The solu ion Yno (Y1)–(Y4) sa is ies he ollowing symme y ela ion: Yn(z) = IN0 0−INYn(¯z)IN0 0−IN,(2.14) which yields some ob ious consequences o he block en ies o Yn. This ela ion is es ablished using he in a iance o (Y1)–(Y4) by such a conjuga ion. Rema k 2.5. Al e na i ely, we can w i e he solu ion (2.8) as Yn(z) = b Pn(z)C(b PnW)(z) −2πiγn−1b Pn−1(z)−2πiγn−1C(b Pn−1W)(z)!, n ∈N,(2.15) whe e b Pndeno e he monic MOPRL o deg ee n o he weigh ma ix W. In he same ein, (Yn)−1(z) =  −2πiC(Wb P∗ n−1)(z)γn−1−C(Wb P∗ n)(z) 2πi b P∗ n−1(z)γn−1b P∗ n(z) , n ∈N.(2.16) Rema k 2.6. Al hough κnis de ined up o a le uni a y ac o , he ma ix coe icien γn in (2.7) is unique. This is a consequence o he uniqueness o he solu ions o he RHP abo e. Rema k 2.7. Fo mulas (2.8) and (2.10) show ha gene ically he beha io o Ynand (Yn)−1 a he endpoin s (and in gene al, any singula poin ) o he suppo o Wis gi en by he local beha io o he Cauchy ans o m o he o hogonali y weigh , also known as i s m- unc ion, see [50,§1.2]. P oo o Theo em 2.2.The ac ha (2.8) o (2.15) is a solu ion o he RHP (Y1)–(Y4) is es ablished ollowing he p oo o he scala case, see e.g. [12], and aking in o accoun ha he o hogonali y o b Pnin (1.1) is equi alen o he homogeneous sys em Zb a xjb Pn(x)W(x)dx =0, j = 0,1, . . . , n −1. The same applies o (2.10) o (2.16) and he RHP (y1)–(y4). Conside he unc ion Zn(z) = Yn(z)yn(z); om (Y2) and (y2) i ollows ha i has no jump ac oss (a, b), and by (Y4), (y4), Zn(z) = O(h(z)), z →c∈ {a, b}. Hence, Znhas only emo able singula i ies a he ini e endpoin s o he suppo o he weigh , and hus is an en i e unc ion. I emains o obse e ha Zn(∞) = I2N o conclude ha Zn(z) = I2N o all z∈C, which p o es ha yn(z) = (Yn(z))−1, as well as he uniqueness o bo h solu ions. The i s iden i y in (2.12) can be es ablished by di ec calcula ion o by obse ing ha his ans o ma ion ca ies he RHP (Y1)–(Y4) o he RHP (y1)–(y4). Finally, he scala unc ion de Yn(z) is analy ic ac oss [a, b], and by (2.8), (2.9), de Yn(z) = de Rn(z) can ha e only emo able singula i ies a he ( ini e) endpoin s o he suppo o W. Hence, de Yn(z) is an en i e unc ion; since by (2.3), de Yn(∞) = 1, we conclude ha i is iden i- cally 1.  8 F.A. G ¨unbaum, M.D. de la Iglesia and A. Ma ´ınez-Finkelsh ein P oposi ion 2.8. Le Pn=κnb Pn,n≥0, be a sequence o o hono mal MOPRL, and le (Qn)n be he co esponding ma ix polynomials o he second kind (2.6). Fo hese polynomials, de ine An=κn−1κ−1 n.(2.17) Then Qn(z)P∗ n−1(z)−Pn(z)Q∗ n−1(z) = A−1 n,(2.18) Qn(z)P∗ n(z) = Pn(z)Q∗ n(z).(2.19) Mo eo e , P∗ n−1(x)AnPn(x)and Q∗ n−1(x)AnQn(x)a e He mi ian o all x∈Rand n≥0. Rema k 2.9. Iden i ies (2.18) and (2.19) a e also known as he Liou ille–Os og adski o mula and he He mi ian p ope y, espec i ely. Bo h we e o iginally de i ed o MOPRL in [20], using a di e en app oach. P oo . Using he explici exp essions o Ynand (Yn)−1(w i en in he o m (2.8) and (2.10)), om block en ies (1,1) and (2,2) o he iden i y Yn(Yn)−1=Iwe ob ain he Liou ille– Os og adski o mula (2.18) while om block en ies (1,2) and (2,1) we ge he so-called He - mi ian p ope y (2.19). Block en ies (1,2) and (2,1) o (Yn)−1Yn=Iyield he commu a i i y ela ions P∗ n−1(z)AnPn(z) = P∗ n(z)A∗ nPn−1(z),Q∗ n−1(z)AnQn(z) = Q∗ n(z)A∗ nQn−1(z), which show ha P∗ n−1(x)AnPn(x) and Q∗ n−1(x)AnQn(x) a e He mi ian o all x∈Rand n≥0.  Fo he o mula ion o he ollowing esul s we need o in oduce a new se o pa ame e s, bn,k =Zb a xkb Pn(x)W(x)dx, k =n, n + 1,..., (2.20) whe e b Pna e he monic MOPRL. In he spi i o [48], we can exp ess hem in e ms o he coe icien s an,j o he polynomials b Pn(see (1.1)) in a o m sui able o nume ical implemen a- ion: Lemma 2.10. Le Ωbe he block lowe iangula ma ix buil up om he coe icien s o he MOPRL (b Pn)n, Ω=     I a1,0I . . .. . .... an,0an,1· · · I     ,so ha     b P0(x) b P1(x) . . . b Pn(x)     =Ω     I xI . . . xnI     ,I=IN. Then bn,k de ined in (2.20)can be ob ained om he las N×Nblock ow o Ω−1as ollows: b∗ n−k,nγn−k=Ω−1n+1,n−k+1, k = 0, . . . , n. (2.21) P oo . Obse e ha b∗ 0,n,...,b∗ n−1,n, b∗ n,n=Zb a xnWb P∗ 0,b P∗ 1,..., b P∗ ndx =Zb a xnW(I, xI, . . . , xnI)Ω∗dx = (µn,µn+1,...,µ2n)Ω∗, P ope ies o Ma ix O hogonal Polynomials ia hei Riemann–Hilbe Cha ac e iza ion 9 whe e µn=Rb axnW(x)dx a e he momen s o he weigh ma ix W. Analogously, by (1.2), (diag (γk)n k=0)−1= diag γ−1 kn k=0 =Zb a     b P0 b P1 . . . b Pn     Wb P∗ 0,b P∗ 1,..., b P∗ ndx =ΩZb a      I xI . . . xnI     WI, xI, . . . , xnIΩ∗=Ω     µ0µ1· · · µn µ1µ2· · · µn+1 . . .. . ..... . . µnµn+1 · · · µ2n     Ω∗. Hence, b∗ 0,nγ0,...,b∗ n−1,nγn−1,b∗ n,nγn =µn,µn+1,...,µ2n     µ0µ1· · · µn µ1µ2· · · µn+1 . . .. . ..... . . µnµn+1 · · · µ2n      −1 Ω−1=0,0,...,IΩ−1, which yields (2.21).  Fo wha ollows i is use ul o single ou he explici exp essions o bn−3,n,...,bn,n, ha a e ob ained om Lemma 2.10 by di ec compu a ions: Co olla y 2.11. Coe icien s bn−k,n, o k= 0,1,2,3, a e gi en by γnbn,n =I, γn−1bn−1,n =−a∗ n,n−1, γn−2bn−2,n =a∗ n−1,n−2a∗ n,n−1−a∗ n,n−2, γn−3bn−3,n =−a∗ n−2,n−3a∗ n−1,n−2a∗ n,n−1+a∗ n−1,n−3a∗ n,n−1+a∗ n−2,n−3a∗ n,n−2−a∗ n,n−3. Now we e u n o Theo em 2.2; as i s immedia e consequence we can ela e he coe icien s in he asymp o ic expansion o Ynand o (Yn)−1, wi h he coe icien s an,j and bn,j: Co olla y 2.12. The coe icien s Yn iin (2.3)a e gi en by Yn i= an,n−i−1 2πibn,n+i−1 −2πiγn−1an−1,n−iγn−1bn−1,n+i−1 , i ≥0.(2.22) Analogously, he coe icien s e Yn iin (2.5)a e gi en by e Yn i=  bT n−1,n+i−1γn−1 1 2πibT n,n+i−1 2πiaT n−1,n−iγn−1aT n,n−i   , i ≥0.(2.23) Rema k 2.13. One o he consequences o (2.14) is ha an,j and bn,j appea ing in (2.22), (2.23) belong o RN×N. The explici exp essions o coe icien s o he asymp o ic expansion abo e, combined wi h he ob ious ac ha Yn(Yn)−1=I2N, yield in a s aigh o wa d way he ollowing iden i ies: 16 F.A. G ¨unbaum, M.D. de la Iglesia and A. Ma ´ınez-Finkelsh ein and o m∈N(see (2.3)), Xn(z) = I2N+ m X i=1 Yn i zi+O(1/zm+1)! znIN0 0z−nIN!V(z), z → ∞.(3.7) Since Xn(z) is in e ible in C R, we can conside he ma ix unc ion Fn(z)=d dz Xn(z)Xn(z)−1. Again, Fn(z) is analy ic in C Rand on he eal line, (Fn)+(x)=(Fn)−(x), which implies ha i is an en i e ma ix unc ion. F om (2.3) and (3.7), o z→ ∞, d dz Yn(z)[Yn(z)]−1=O1 z, and combining i wi h (2.3), (2.5), (3.5) and (3.7), we ge o z→ ∞, Fn(z) = d dz Xn(z)Xn(z)−1 = I2N+ m X i=1 Yn i zi+O1/zm+1! G(z)0 0−G∗(z)! × I2N+ m X i=1 e Yn i zi+O1/zm+1!+O(1/z).(3.8) By Liou ille’s heo em, he igh hand side in (3.8) will coincide wi h i s polynomial pa in he expansion a in ini y, and i s deg ee is no g ea e han he deg ee o G. To be mo e p ecise, i we assume ha he deg ee o Gis m∈Nand deno e G(z) = m X j=0 Mjzj, Mj= Mj0 0−M∗ j!, hen a e d opping he nega i e powe s o zin Fn(z) = I2N+ m X i=1 Yn i zi+O1/zm+1! ×  m X j=0 Mjzj  I2N+ m X k=1 e Yn k zk+O1/zm+1!+O(1/z) we ob ain ha Fn(z) = m X k=0   m X j=k j−k X i=0 Yn i Mje Yn j−i−k zk,wi h Yn 0=e Yn 0=I2N.(3.9) Fo ins ance, i m= 0, Fn(z) = M0,(3.10) o m= 1, Fn(z) = M1z+ M0+Yn 1 M1+ M1e Yn 1,(3.11) and o m= 2, Fn(z) = M2z2+ ( M1+Yn 1 M2+ M2e Yn 1)z+ M0+Yn 1 M1+ M1e Yn 1 +Yn 2 M2+Yn 1 M2e Yn 1+ M2e Yn 2.(3.12) Finally, using he explici exp essions (2.22) and (2.23) in (3.9), we a i e a he ollowing P ope ies o Ma ix O hogonal Polynomials ia hei Riemann–Hilbe Cha ac e iza ion 17 Theo em 3.1. Unde assump ions (3.2)and (3.4), wi h G(z) = m P j=0 Mjzj∈Pm, he ma ix unc ion Xnde ined in (3.6)sa is ies he ollowing i s -o de di e en ial equa ion wi h polyno- mial coe icien s: d dz Xn(z) = Fn(z;G)Xn(z),(3.13) whe e Fn(z;G) =  −Bn(z;G)−1 2πiγ−1 nAn(z;G) 2πi An−1(z;G)γn−1B∗ n(z;G) ,(3.14) Anand Bna e ma ix polynomials, An(z;G) = −γn  m−1 X j=0 bn,n+m−j−1∆∗ j,n(z) + ∆j,n(z)b∗ n,n+m−j−1 ,(3.15) Bn(z;G) = −  m X j=0 ∆j,n(z)b∗ n−1,n+m−j−1+bn,n+m−j−2∆∗ j,n−1(z) γn−1,(3.16) and he coe icien s ∆j,n(z)a e gi en by ∆j,n(z) = j X k=0 b Pn,k(z)Mm−j+k,b Pn,k(z) = zkI+an,n−1zk−1+· · · +an,n−k. Mo eo e , Fn(·;G)is linea in G: Fn(z;G1+G2) = Fn(z;G1) + Fn(z;G2).(3.17) Rema k 3.2. The coe icien s bn,k we e in oduced in (2.20) and discussed in Lemma 2.10. I is wo h obse ing he simila i y o his esul wi h Theo em 2.17; in he p esen si ua ion we can compu e he di e en ial equa ion o Xndi ec ly in e ms o he coe icien s o he MOPRL b Pn wi hou conside ing in eg als o s udying he beha io a he endpoin s. Finally, we ha e once again ha γnA∗ n(z;G) = An(z;G)γn.(3.18) Again, o lowes deg ees min Theo em 3.1 we can use (3.10), (3.11), (3.12), Lemma 2.10 and Co olla y 2.11 in o de o w i e he coe icien ma ix (3.14) o he di e en ial equa ion explici ly. Fo ins ance, o m= 0, Fn(z;G) = M00 0−M∗ 0; o m= 1, Fn(z;G) = G(z) + an,n−1M1−M1an,n−1 1 2πi(γ−1 nM∗ 1+M1γ−1 n) −2πi(γn−1M1+M∗ 1γn−1)−G∗(z) + a∗ n,n−1M∗ 1−M∗ 1a∗ n,n−1 ,(3.19) and o m= 2, wi h he no a ion (3.14), Bn(z;G) = −G(z)−(an,n−1M2−M2an,n−1)z+M1an,n−1−an,n−1M1−an,n−2M2 18 F.A. G ¨unbaum, M.D. de la Iglesia and A. Ma ´ınez-Finkelsh ein −M2(an+1,nan,n−1+an+1,n−1) + an,n−1M2an,n−1−γ−1 nM∗ 2γn−1, An(z;G) = −M∗ 2z−M∗ 1+a∗ n+1,nM∗ 2−M∗ 2a∗ n,n−1 −γn(M2z+M1−M2an+1,n +an,n−1M2)γ−1 n. Obse e om Theo em 3.1 ha Bn(z;G) is a ma ix polynomial o deg ee a mos m and An(z;G) is a ma ix polynomial o deg ee a mos m−1. Finally, i is impo an o emphasize ha in many si ua ions we can exploi he implica ions o he eedom in he ac o iza ion (3.3) on he di e en ial equa ion (3.13). This eedom, as we will see in P oposi ion 3.9, appea s only in he ma ix se ing. P oposi ion 3.3. I unde assump ions (3.2)and (3.4), wi h Ga polynomial, he e exis s a non- i ial ma ix- alued unc ion S, non-singula on C, smoo h and uni a y on R, such ha H(z) = T(z)S0(z)S∗(z)T−1(z) (3.20) is also a polynomial, hen e T=T S sa is ies W(x) = e T(x)e T∗(x), x ∈R,e T0(z) = e G(z)e T(z), z ∈C, wi h e G(z) = G(z) + H(z). Mo eo e , he ma ix Xnde ined in (3.6), sa is ies along wi h (3.13)–(3.16) he ollowing ela ion: d dz Xn(z) = (Fn(z;G) + Fn(z;H)) Xn(z)−Xn(z) χ(z)0 0−χ∗(z)!,(3.21) wi h χ(z) = S0(z)S∗(z). Rema k 3.4. Equa ions (3.13) and (3.21) a e no necessa ily i ially ela ed, and in p inciple we could combine hem in o de o es ablish new ela ions o Xn, and hus, o Yn. P oo . The key obse a ion is he o mula (3.17), so ha he coe icien s Anand Bnin (3.13) depend linea ly on G: An(·;G+H) = An(·;G) + An(·;H),Bn(·;G+H) = Bn(·;G) + Bn(·;H).(3.22)  3.2 Di e en ial p ope ies Taking in o accoun he simila i ies be ween Theo ems 2.17 and 3.1, we ge immedia ely he analogue o P oposi ion 2.20 ( he compa ibili y condi ions): P oposi ion 3.5. Unde assump ions (3.2)and (3.4), wi h G(z)a polynomial, he coe icien s o he di e en ial equa ion (3.13)sa is y he ollowing ecu ence ela ions: o e e y n≥0, I+Bn+1(z;G)(zI−αn)−(zI−αn)Bn(z;G) = A∗ n+1(z;G)βn+1 −βnA∗ n−1(z;G) (3.23) and Bn+1(z;G) + γ−1 nB∗ n(z;G)γn= (zI−αn)A∗ n(z;G),(3.24) whe e αnand βna e he coe icien s o he h ee- e m ecu ence ela ion (2.30). As be o e, he ladde ope a o s ( he mos basic di e en ial p ope ies o MOPRL) can be easily ob ained by analyzing he i s block column o Xnin he di e en ial equa ion (3.13): P ope ies o Ma ix O hogonal Polynomials ia hei Riemann–Hilbe Cha ac e iza ion 19 P oposi ion 3.6. Unde assump ions (3.2)and (3.4), wi h G(z)a ma ix polynomial, he monic MOPRL (b Pn)nsa is y he ollowing di e ence-di e en ial ela ions (lowe ing and aising ope a o s, espec i ely): b P0 n(z) + b Pn(z)G(z) = −Bn(z;G)b Pn(z) + A∗ n(z;G)βnb Pn−1(z) (3.25) and b P0 n(z) + b Pn(z)G(z) = A∗ n(z;G)(zI−αn)− Bn(z;G)b Pn(z)− A∗ n(z;G)b Pn+1(z),(3.26) whe e Anand Bna e gi en by (3.15)and (3.16), espec i ely. P oo . Block en y (1,1) o (3.13) gi es he lowe ing ope a o (3.25), using (2.31) and (3.18), while block en y (2,1) o (3.13) gi es he aising ope a o (3.26) using (3.24) and (3.18).  Rema k 3.7. Compa ing wi h he esul s o Co olla y 2.21, no ice ha hese ladde ope a o s con ain a e m wi h he MOPRL mul iplied on he le . In he si ua ion desc ibed in P oposi ion 3.3, we can in p inciple exploi he non-uniqueness o he ela ions abo e in o de o de i e u he ela ions o he MOPRL. The ollowing esul shows he possibili y o he exis ence o 2- e ms ecu ence ela ions o he amily b Pn, a phenomenon ha has no been epo ed be o e in he heo y o MOPRL. Co olla y 3.8. Unde condi ions o P oposi ion 3.3, he amily o monic MOPRL (b Pn)nsa is- ies, along wi h (3.25)–(3.26), he ollowing ela ions: b Pn(z)H(z) = −Bn(z;H)b Pn(z) + A∗ n(z;H)βnb Pn−1(z) (3.27) and b Pn(z)H(z) = A∗ n(z;H)(zI−αn)− Bn(z;H)b Pn(z)− A∗ n(z;H)b Pn+1(z),(3.28) whe e His de ined in (3.20). In pa icula , p o ided An(z;G)is in e ible o all z∈C, b Pn(z)H(z) + Bn(z;H)b Pn(z)− A∗ n(z;H)A−∗ n(z;G) ×b P0 n(z) + b Pn(z)G(z) + Bn(z;G)b Pn(z)=0.(3.29) P oo . To ge (3.27), we sub ac (3.13) and (3.21) and hen e alua e he (1,1) block en y. (3.28) is ob ained om (3.27) and he h ee e m ecu ence ela ion. Finally, (3.29) is ob ained by eplacing (3.25) in (3.27).  Equa ions (3.27) and (3.28) a e known as he 0- h o de ladde ope a o s, while (3.29) is a i s o de di e en ial ela ion o he MOPRL. In some si ua ions hey yield i ial iden i ies; his is always ue in he scala case, as he ollowing p oposi ion shows: P oposi ion 3.9. Assume ha unde he condi ions o P oposi ion 3.3,χ(z) = S0(z)S∗(z) = ip(z)I, whe e pis a scala polynomial o deg ee m. Then An(z;H) = 0,and Bn(z;H) = −ip(z)I. P oo . F om (3.20), H=T χT −1, and by linea i y o Anand Bnin H(see (3.22)), i is enough o p o e he o mulas abo e o monomials zkI. In his case, o mulas (3.15) and (3.16) simpli y conside ably. Using P oposi ion 2.14 gi es An(z;H) = 0and Bn(z;H) = −izkI. 20 F.A. G ¨unbaum, M.D. de la Iglesia and A. Ma ´ınez-Finkelsh ein In he scala case (N= 1) he only smoo h and uni a y unc ion s(x) on Rhas he o m s(z) = eip(x), wi h p eal- alued, and s0(z)s(¯z) = ip(z), so ha he assump ion ha pis a polynomial b ings us o he si ua ion desc ibed in P opo- si ion 3.9. This explains why in he scala se ing we ne e ge non i ial 0- h o de ladde ope a o s. In he scala case one canno ha e i s o de di e en ial ela ions o he MOPRL, such as hose gi en in (3.29). The gene al ma ix case is much iche and complex, as will be illus a ed wi h some examples in he nex sec ion. We inish his sec ion wi h a class o second-o de di e en ial equa ions sa is ied by he MOPRL ( b Pn)n. As a consequence o he P oposi ion 3.6 we ha e he ollowing P oposi ion 3.10. Unde assump ions (3.2)and (3.4), wi h G(z)a polynomial, he MOPRL (b Pn)nsa is y he ollowing second-o de di e en ial equa ion b P00 n+ 2 b P0 nG+b PnG0+G2+Mnb P0 n+Nnb Pn+Mnb PnG=0,(3.30) whe e An(z) = An(z;G),Bn(z) = Bn(z;G), Mn(z) = Mn(z;G) =−(A∗ n(z))0A−∗ n(z) + Bn(z)− A∗ n(z)(zI−αn) + A∗ n(z)Bn+1(z)A−∗ n(z), and Nn(z) = Nn(z;G) = Mn(z)Bn(z)− B2 n(z) + B0 n(z) + A∗ n(z)βnA∗ n−1(z), p o ided ha he in e se o An(z;G)exis s o z∈C. He e, again, −∗ deno es he conjuga e anspose o he in e se. P oo . Di e en ia e he lowe ing ope a o (3.25) and subs i u e he aising ope a o (3.26) e alua ed a n−1.  Rema k 3.11. No e ha we can easily ob ain ano he second-o de di e en ial ope a o sa - is ied by he o hogonal polynomials e e sing he ladde ope a o s. This new di e en ial equa- ion needs no be in p inciple he same as (3.30). Ne e heless, i is s aigh o wa d o see ha bo h equa ions a e equi alen using he compa ibili y condi ions (3.23) and (3.24). The equa ion (3.30) does no ha e he o m o he igh hand side di e en ial ope a o conside ed o ins ance in [23], due o he e ms Mnb P0 n,Nnb Pnand Mnb PnG. In some cases, unde addi ional assump ions on he weigh , (3.30) can be educed u he , as we will see in he ollowing sec ion. 4 Illus a i e examples In his sec ion we s udy a numbe o examples o weigh s Wwhich a e smoo h and non- anishing on he whole eal line; his assump ion simpli ies he Riemann–Hilbe o mula ion because in his case one does no conside he local condi ions (Y4) (see Sec ion 2.1). Addi ionally, wi hou loss o gene ali y, we ake W(0) = I. Fo con enience, we conside he weigh s o he o m W(x) = e−2q(x)U(x)U∗(x), x ∈R,(4.1) P ope ies o Ma ix O hogonal Polynomials ia hei Riemann–Hilbe Cha ac e iza ion 21 whe e qis a scala eal- alued unc ion, so ha wi h he no a ion (3.2) and (3.4), we may ake T(x) = e−q(x)U(x) and G(x) = −q0(x)I+U0(x)U−1(x).(4.2) We a e in e es ed in he case when Gis a ma ix polynomial. Hence, keeping up wi h p e ious hypo heses, we will assume ha qis a (scala ) polynomial o e en deg ee wi h eal coe icien s and a posi i e leading coe icien , and ha U0(x)U−1(x) is a ma ix polynomial. Since ou main goal he e is o gene a e a se o examples (some new, some al eady well known), we es ic he deg ee o q o ei he 2 ( he He mi e case) o 4 ( he F eud case), and he deg ee o U0(x)U−1(x) o a mos 1. We s a by conside ing he case when U0(x)U−1(x) is a monomial, i.e. ei he U0(x)U−1(x) = Ao U0(x)U−1(x)=2Bx o cons an ma ices A,B∈CN×N. Taking in o accoun he linea i y (3.22) o he coe icien s Anand Bnin (3.15), (3.16), we can ob ain he di e en ial equa ion (3.13) o he gene al case o U0(x)U−1(x) = A+ 2Bx. Howe e , inding he co esponding o hogonali y weigh is mo e in ol ed: when Aand Bdo no commu e, sol ing U0(x)U−1(x) = A+ 2Bxis no s aigh o wa d. In Sec ion 4.1.3 we will discuss some examples, which yield explici exp essions o U(x), ela ed o his case. Recen ly an example has been ound in [3] whe e a weigh ma ix suppo ed in he eal line is explici ly gi en (bu no o he ype (4.1)), when G(x) is a ma ix polynomial o deg ee N wi h in gene al non-commu ing coe icien s. 4.1 The He mi e case Fo q(x) = x2/2, le us conside wo cases, i s U0(x)U−1(x) = Aand hen U0(x)U−1(x) = 2Bx. We end up his Sec ion by discussing b ie ly he case o U0(x)U−1(x) = A+ 2Bx. 4.1.1 U0(x)U−1(x) = A The di e en ial equa ion U0(x)U−1(x) = A, so ha U(x) = eAx,T(x) = e−x2/2eAx, and he weigh ma ix (4.1) is gi en by W(x) = e−x2eAxeA∗x,A∈CN×N, x ∈R.(4.3) The ma ix Ghas he o m G(x) = −xI+A, and acco ding o (3.19), An(x;G) = 2I,Bn(x;G) = −G(x). The compa ibili y condi ions o P oposi ion 3.5 yield 2(βn+1 −βn) = I+Aαn−αnA,(4.4) and αn=1 2(A+γ−1 nA∗γn).(4.5) The lowe ing and aising ope a o s om P oposi ion 3.6 a e educed now o b P0 n(x) + b Pn(x)A−Ab Pn(x) = 2βnb Pn−1(x),(4.6) and −b P0 n(x)+2xb Pn(x) + Ab Pn(x)−b Pn(x)A−2αnb Pn(x) = 2 b Pn+1(x). 22 F.A. G ¨unbaum, M.D. de la Iglesia and A. Ma ´ınez-Finkelsh ein Summing up he elescopic ela ion in (4.4) (o compa ing he O(xn−1) e m in (4.6)) gi es βn=1 2(nI+an,n−1A−Aan,n−1).(4.7) Wi h he no a ion o P oposi ion 3.10, Mn(x) = 2 (αn−A),Nn(x) = −2 (αn−A)G(x)−G2(x) + I+ 4βn, so ha he di e en ial equa ion (3.30) o he monic polynomials b Pn, o hogonal wi h espec o he weigh (4.3), boils down o b P00 n(x)+2b P0 n(x)(A−xI) + b Pn(x)(A2−2xA) = (−2xA+A2−4βn)b Pn(x) + 2(A−αn)( b P0 n(x) + b Pn(x)A−Ab Pn(x)).(4.8) These o mulas hold o any cons an ma ix A, and we canno expec impo an sim- pli ica ions wi hou na owing he class o he weigh s u he . This can be done assuming in addi ion ha he hypo heses o P oposi ion 3.3 hold. This, as i was shown in [23,24], imposes addi ional cons ain s on he weigh W. Since he cons uc ion is desc ibed in de ail in [23,24], he exposi ion in his pa will be a he ske chy. The ma ix H om (3.20), es ic ed o R, can be w i en as H(x) = T(x)χT −1(x) = eAxχe−Ax =χ+ adA(χ)x+ ad2 A(χ)x2 2+· · · =X k≥0 adk A(χ)xk k!.(4.9) He e χ(x) = S0(x)S∗(x) is skew-He mi ian on R, adAis he commu a o gi en by adA(χ) = Aχ −χA, and we de ine ecu si ely ad0 A(χ) = χ,adn+1 A(χ) = adA(adn A(χ)) o n≥1. The simples si ua ion ob ains when he igh hand side in (4.9) is cons an ; as i was shown in Lemma 2.4 o [24], his assump ion yields χ=iaI o ce ain a∈R(which is he case discussed in P oposi ion 3.9), when he e a e no new ladde ope a o s. Consequen ly, o he i s non- i ial si ua ion we mus assume ha (4.9) is a ma ix polynomial o deg ee a leas one. We conside wo si ua ions ha yield deg ee exac ly one (see [23,24] o mo i a ions and u he de ails). Le us de ine a nilpo en ma ix o he o m L= N−1 X k=1 νkEk,k+1, νk∈C {0},(4.10) whe e Eij is a ma ix wi h 1 a en y (i, j) and 0 elsewhe e, and a diagonal ma ix J= N X k=1 (N−k)Ek,k.(4.11) Fo he i s non- i ial example we assume ha A=Land χ=iJ, so ha adA(χ) = −A(4.12) and ad2 A(χ) = 0. Thus, H(x) = eAxχe−Ax=i(J−Ax), and by (3.19), An(x;H) = i(−A∗+γnAγ−1 n)=2i(α∗ n−A∗), P ope ies o Ma ix O hogonal Polynomials ia hei Riemann–Hilbe Cha ac e iza ion 23 Bn(x;H) = i(Ax−J+an,n−1A−Aan,n−1) = i(Ax−J+ 2βn−nI). F om he compa ibili y condi ions o P oposi ion 3.5 we ge Jαn−αnJ+αn=A+1 2A2αn−αnA2and J−γ−1 nJγn=Aαn+αnA−2α2 n,(4.13) whe e we ha e used ela ions (4.4), (4.5), (4.7) and (A−2αn)βn=βn(A−2αn−1), which can be easily p o ed using (2.31) and (4.5). Thus, om Co olla y 3.8, (4.4) and (4.13), he lowe ing and aising ope a o s (o he 0- h o de ) a e b Pn(x)J−Jb Pn(x)−x(b Pn(x)A−Ab Pn(x)) + 2βnb Pn(x)−nb Pn(x) = 2(A−αn)βnb Pn−1(x), and b Pn(x)(J−xA)−γ−1 n(J−xA∗)γnb Pn(x)+2βn+1 b Pn(x)−(n+ 1) b Pn(x) = 2(αn−A)b Pn+1(x), espec i ely, which yields by (3.29) he i s o de ela ion (A−αn)b P0 n(x)+(A−αn+xI)( b Pn(x)A−Ab Pn(x)) −2βnb Pn(x) =b Pn(x)J−Jb Pn(x)−nb Pn(x).(4.14) These iden i ies hold in addi ion o he second o de equa ion ob ained in (4.8), alid as we ecall, o any A. As a as we a e awa e o , hese ela ions a e new. We use hem o simpli y he di e en ial equa ion ema kably: mul iplying (4.14) by 2 and plugging i in o (4.8) gi es us o A=L, b P00 n(x)+2b P0 n(x)(A−xI) + b Pn(x)A2−2J=−2nI+A2−2Jb Pn(x), which is a linea second-o de di e en ial equa ion wi h coe icien s in he igh hand side independen on n(a.k.a. S u m–Liou ille equa ion wi h polynomial coe icien s), s udied by Du ´an and G ¨unbaum in [23]. In he second case we ake A=L(I+L)−1= N−1 X j=1 (−1)j−1Lj,χ=iJ, whe e Lis gi en in (4.10) and Jin (4.11). Consequen ly, adA(χ) = −A+A2,ad2 A(χ) = 0,(4.15) and H(x) = eAxχe−Ax=iJ−A−A2x. In his case, An(x;H) = i−A∗+ (A∗)2+γnA−A2γ−1 n 24 F.A. G ¨unbaum, M.D. de la Iglesia and A. Ma ´ınez-Finkelsh ein = 2i(α∗ n−A∗−(α∗ n−A∗)α∗ n−α∗ n(α∗ n−A∗)) and Bn(x;H) = iA−A2x−J+an,n−1A−A2−A−A2an,n−1 =iA−A2x−J+ 2βn−nI−2(Aβn+βnA)+2nA. As o new esul s, we ge he lowe ing ope a o , b Pn(x)J−xA−A2−J−xA−A2b Pn(x) = (−2βn+n(I−2A) + 2(Aβn+βnA)) b Pn(x) −2(αn−A−(αn−A)αn−αn(αn−A))βnb Pn−1(x), he aising ope a o b Pn(x)J−xA−A2−J−xA−A2b Pn(x) = (−2βn+n(I−2A) + 2(Aβn+βnA)) b Pn(x) −2(αn−A−(αn−A)αn−αn(αn−A))((xI−αn)b Pn(x)−b Pn+1(x)), and he i s -o de di e en ial equa ion (3.29): (A−αn−(A−αn)αn−αn(A−αn)) b P0 n(x) =b Pn(x)J−Jb Pn(x)−xb Pn(x)A2−A2b Pn(x) + (xI−A+αn+ (A−αn)αn+αn(A−αn))( b Pn(x)A−Ab Pn(x)) + (2βn+n(2A−I)−2(Aβn+βnA)) b Pn(x). Mul iplying his equa ion by 2 and plugging i in o he second-o de di e en ial equa ion (4.8) gi es b P00 n(x)+2b P0 n(x)(A−xI) + b Pn(x)A2−2xA2−2J=A2−2xA2−2Jb Pn(x) + (2n(2A−I)−4(Aβn+βnA) + 2((αn−A)αn+αn(αn−A))A)b Pn(x) −2((αn−A)αn+αn(αn−A))( b P0 n(x) + b Pn(x)A). This di e en ial equa ion is no o S u m–Liou ille ype conside ed by Du ´an and G ¨unbaum in [23], bu ne e heless we ha e been able o gi e a numbe o di e en ial equa ions o i s and second o de sa is ied by MOPRL wi h espec o a weigh ma ix ha ha e no been conside ed up o his poin . 4.1.2 U0(x)U−1(x)=2Bx The weigh ma ix (4.1) is now gi en by W(x) = e−x2eBx2eB∗x2,B∈CN×N, x ∈R, whe e we assume ha Bis chosen such ha all he momen s exis . By (4.2), T(x) = e−x2/2eBx2and G(x) = (2B−I)x. The weigh ma ix is an e en ma ix unc ion, so ha all momen s o odd o de anish. As a consequence, an,n−1=0and αn=0. Then An(x;G) = 2I−B∗−γnBγ−1 n,Bn(x;G)=(I−2B)x, P ope ies o Ma ix O hogonal Polynomials ia hei Riemann–Hilbe Cha ac e iza ion 25 and he e will be only one compa ibili y condi ion: 2I−B−γ−1 n+1B∗γn+1βn+1 −2βnI−B−γ−1 n−1B∗γn−1=I. F om (4.16) we ge he explici exp ession o βn ia 2I−B−γ−1 nB∗γnβn=nI+ 2(an,n−2B−Ban,n−2). The lowe ing and aising ope a o s a e b P0 n(x)+2x(b Pn(x)B−Bb Pn(x)) = 2I−B−γ−1 nB∗γnβnb Pn−1(x),(4.16) and b P0 n(x)+2xb Pn(x)B−Bb Pn(x)= 2I−B−γ−1 nB∗γnxb Pn(x)−b Pn+1(x), espec i ely. This yields he ollowing second-o de di e en ial equa ion b P00 n(x)+2xb P0 n(x)(2B−I)+2xγ−1 nB∗γn−Lnb P0 n(x)+4x2b Pn(x)B2−B2b Pn(x) + 4Knb Pn(x) + 2−4x2I+ 4x2γ−1 nB∗γn−Lnb Pn(x)B−Bb Pn(x)=0, whe e Ln=I−B−γ−1 nB∗γnBI−B−γ−1 nB∗γn−1, and Kn=I−B−γ−1 nB∗γnβnI−B−γ−1 n−1B∗γn−1. These exp essions a e alid o any B(as long as all momen s o he weigh ma ix exis ). Fo u he simpli ica ions we can assume again ha he hypo heses o P oposi ion 3.3 hold, and conside he cases gi en by he algeb aic ela ions (4.12) and (4.15). Fo ins ance, when (4.15) holds, we ob ain again a second-o de di e en ial equa ion wi h coe icien s in he igh hand side independen on n, s udied by Du ´an and G ¨unbaum in [23]. The compu a ions a e simila and will be omi ed o he sake o b e i y. 4.1.3 O he cases Assume now ha we ha e a linea combina ion o he p e ious wo cases, i.e. U0(x)U−1(x) = A+ 2Bx, A,B∈CN×N,wi h U(0) = I,(4.17) in which case he ma ix G(x) is gi en by G(x) = A+ (2B−I)x. This is all ha is needed o calcula e he coe icien s An(x;G) and Bn(x;G) and consequen ly he compa ibili y condi ions, he ladde ope a o s and he di e en ial ela ions. Howe e , o he co esponding o hogonali y weigh we need o sol e (4.17) explici ly, which may be non- i ial (unless Aand Bcommu e). In gene al, his solu ion can be gi en in e ms o he ime o de ed exponen ial U(x) = : eRx 0(A+Bs)ds :,A,B∈CN×N, x ∈R. This is ob ained by ew i ing he di e en ial equa ion as an in eg al one and “sol ing” i by i e a ion. In gene al, one canno gi e an explici exp ession o his in ini e sum. An example o explici ly sol able non- i ial equa ion (4.17) can be ound in [22, Theo em 1.1], whe e non-commu ing Aand Ba e cons uc ed (as a linea combina ion o he ma ices Land J