Can. J. Math., Vol. XXXVIII, No. 2, 1986, pp. 397-415 ON SIEVED ORTHOGONAL POLYNOMIALS II: RANDOM WALK POLYNOMIALS JAIRO CHARRIS AND MOURAD E. H. ISMAIL 1. Introduction. A birth and death process is a stationary Markov process whose states are the nonnegative integers and the transition probabilities (1.1) pmn(t) = Pr{X(t) = n\X(0) = m) satisfy Pmt + 0(/) n = m + 1 (1.2) Pm„(t) = 8mt + 0(0 n = m - 1 1 — 08m + Sm)t + o(0 n = m, as / —> 0. Here we assume /?n > 0, 8n + 1 > 0, « = 0, 1,. , but ô0 ^ 0. Karlin and McGregor [10], [11], [12], showed that each birth and death process gives rise to two sets of orthogonal polynomials. The first is the set of birth and death process polynomials {Qn(x) } generated by Ôo(*) = 1> Gi(*) = (A) + «b " ^Vi8o, « > 0. In this case there exists a positive measure da supported on [0, oo) such that oo Qn(x)Qm(x)da(x) = 8mn/iTn, m, n = 0, 1,. /; o holds where •un = /yj,... p„-l/{sls2... 8„), « > o, «o = l. The second set is the set of random walk polynomials. They arise when one studies a random walk on the state space. The random walk polynomials {Rn(x) } satisfy the recursion (1.3) xRn(x) = BnRn + x(x) + AA-i(*)> n > 0 and the initial conditions Received October 3, 1984 and in revised form February 14, 1985. This research was partially supported by NSF Grant MCS 8313931, Arizona State University and the National University of Colombia. 397 Downloaded from https://www.cambridge.org/core. 03 Oct 2021 at 03:27:13, subject to the Cambridge Core terms of use. 398 J. CHARRIS AND M. E. H. ISMAIL (1.4) R0(x) = 1, /?,(*) = X/BQ, with (1.5) Bn = B„/(Bn + 8n), Dn = 8n/(B„ + 8„). Clearly Bn + Dn = 1. The random walk polynomials are orthogonal with respect to a positive measure supported in [— 1, 1]. In fact (1.6) /-.*» ,{x)Rn(x)dR(x) = A„8„ where (1.7) A0 = l, An = {DlD2...Dn}/{B0Bl...B„_]), n > 0. The ultraspherical (Gegenbauer) polynomials {Cn(x) } are random walk polynomials with l l (1.8) Bn = -(n + \)/(n XI Dn = -(n + 2X \)/(n + A), ( [15] and [17] ). On the other hand the random walk polynomials associated with (1.9) Bn = ±(n + 2X)/(n X), Dn = -n/(n + A), k are {n\C n(x)/(2X)n}, where (1.10) (a)0 = 1, (a)w = a(o + 1) (a + « - 1), « > 0. Al-Salam, Allaway and Askey [1] observed that two limiting cases of the Rogers continuous ^-ultraspherical polynomials, [2], [4], are interesting. In both cases q approached exp(2<ni/k), k is a given positive integer, k > 1. This led them to define the sieved ultraspherical polynomials of the first kind by x c 0(x; k) 1, c\x; k) = x, (m + 2\)cA (x; k) = 2x(m + \)cA (x; k) (l.n) mk+l mk ~ »»<£,*_i(x; k), m > 0, x 4+1(x; k) = 2xc n(x; k) - c^_,(x; k), k{ n, n > 0 and the sieved ultraspherical polynomials of the second kind via B^ix; k) = 1, Bx(x; k) = 2x, x mB mk(x; k) = 2x(m + A)^_,(x; k) (1.12) x - (m + 2\)B mkJx; k), m > 0 ££+,(*; it) = 2xBx(x; k) - 2**_,(;c; it), A: j n + 1, n > 0. In this work, we generalize the sieved ultraspherical polynomials to a fairly large class of random walk polynomials. This is done by starting Downloaded from https://www.cambridge.org/core. 03 Oct 2021 at 03:27:13, subject to the Cambridge Core terms of use. RANDOM WALK POLYNOMINALS 399 with a set of random walk polynomials {Rn(x) } satisfying (1.3) and (1.4). We shall also assume that k > 1 is a given integer and (1.13) Bn + Dn = l9 0<Bn< In = 0, 1, . The sieved random walk polynomials of the first kind are generated by (1.14) r0(x) = 1, rx(x) = x, xrn(x) = dn_xrn + x(x) + bn_xrn_x(x), n > 0, while the sieved random walk polynomials of the second kind are defined recursively by (1.15) s0(x) = 1, sx(x) = 2x, xsn(x) = lOO + dnSn-\(X\ n > °> where l (1.16) bn = dn= - ilk\n+\,bnk_x=Bn^,dnk_x=Dn_x. In particular, when Bn, Dn are defined by (1.9), Rn(x) = Cn(x), the rw's 9 and sn's are essentially the cn s and ^'s of Al-Salam, Allaway and Askey. We shall establish explicit formulas and generating functions for rn(x) and sn(x) in terms of Rn(x) and the Chebyshev polynomials, see (2.3), (2.5), (2.6), (3.4) and (3.5). In Section 4 we shall show that such formulas hold only for random walk polynomials. In Section 5 we shall show how to compute the Stieltjes transform of the distribution (spectral) function of {rn(x) } from the asymptotics of the random walk polynomials {Rn(x) } and their duals {Sn(x) }. The dual polynomials {Sn(x) } are the random walk polynomials (1.17) S0(x) = 1, S{(x) = x/DQ, xSn(x) = DnSn + x(x) + BnS„^(x). Karlin and McGregor [11] studied random walk polynomials {Rn(x) } when 8n = n and (in = b. Carlitz [6] was generalizing earlier work of Tricomi and independently discovered the same set of polynomials at the same time. Chihara [7] calls them the Tricomi-Carlitz polynomials but we shall call them the Carlitz-Karlin-McGregor (CKM) polynomials and denote them by {rn(x\ b) }, as in [3]. They are recursively defined by ; ^" \ x(n + b)rn(x; b) = brn + x{x\ b) + nrn_x(x\ b), n ^ 0. Carlitz proved their orthogonality using Euler's identity (1.19) é° = 1 + a 2 (a H) (ze-y, n = \ n\ Downloaded from https://www.cambridge.org/core. 03 Oct 2021 at 03:27:13, subject to the Cambridge Core terms of use. 400 J. CHARRIS AND M. E. H. ISMAIL but Karlin and McGregor used probabilistic methods to compute their distribution function. Their orthogonality relation is oo oo (1.20) 2 °,rm(x-; b)rn(x:\ b) + 2 °fm(-xj9 b)rn(-x; b) = hn8 where b l X n (1.21) a- = -^(b +jy~ exp(b +j), hn = n\b - /(b + n). These remarkable polynomials are discrete analogues of the Hermite polynomials and their distribution function is a step function (with infinitely many steps) that approximates the integral x 9 / exp(-rW/. To see this let nl (1.22) qn(x) = {lb) \{x^Vb\ b\ It is easy to see that q0(x) = 1, qx(x) = 2x and 2x(\ 4- n/b)qn(x) = qn + \(x) + 2nqn__x(x\ which when compared with H0(x) = 1, Hx(x) = 2x9 2xHn(x) = Hn + X{x) + 2nHn^(x)9 [15, page 188], shows that lim qn(x) = Hn(x)9 b-*oo hence nll (1.23) lim {2b) rn{x^/2lb\ b) = Hn(x). b^oo In Section 6 a sieved analogue of the CKM polynomials will be introduced. We apply the results of Section 3 and 5 to obtain explicit formulas and generating functions for the sieved CKM polynomials. We then apply Theorem 5.1 and compute the distribution function of the sieved CKM polynomials. 2. Sieved polynomials of the second kind. Recall that these polynomials satisfy (1.15) and (1.16). We shall adopt the convention (2.1) £/_,(*) = R_}(x) = 0. The elementary trigonometric identity (2.2) Un(x) = Un_2(x) + 2Tn(x) Downloaded from https://www.cambridge.org/core. 03 Oct 2021 at 03:27:13, subject to the Cambridge Core terms of use. RANDOM WALK POLYNOMINALS 401 will be used repeatedly. We now prove: THEOREM 2.1. The explicit representations (2.3) snk+l(x) = U^R^T^x)) + Uk_l_2(x)Rn_l(Tk(x)X hold for I = 0, 1, . ., k - 1, n = 0, 1,. Proof. Let snk+l(x) denote the right side of (2.3). These sn's clearly satisfy the initial conditions in (1.15) so it remains to show that the polynomials also satisfy the recursion in (1.15). It is straightforward to obtain the recursion 2xsnk+l{x) = snk+l+x(x) + snk+t_x(x\ I = 0, 1, .. ., k - 2, from the recurrence relation (2.4) 2xUn(x) = £/„+,(*) + £/„_,(*)• This proves the recursion in (1.15) when n = mk + /, / = 1, 2, . , k — 2. The case 1 = 0 can be similarly proved since (2.4) holds for n = 0 and Sl(x) = £//(*), / = 0, 1, ...,& - 1. The case / = k — 1 can be proved as follows. First observe that 2xsnk+k_x(x) = 2xUk_x(x)Rn(Tk{x)) = {Uk(x) + Uk_2(x)}Rn(Tk(x)) = 2{Tk(x)+ Uk_2(x)}Rn(Tk(x)), in view of (2.2) and (2.4). Now (1.3) and the above relationship yield xsnk+k_x{x) = BnRn+x{Tk(x)) + DnRn_x{Tk{x)) + Uk_2(x)Rn(Tk(x)) = Bn{Rn+x(Tk(x)) + Uk_2(x)Rn(Tk(x))} + D„{Uk_.2(x)R„(Tk(x)) + R„^(Tk(x))}, where we used (1.13). Thus (2.3) holds when n = mk + k — 1. This identifies the right sides of (2.3) as the polynomials under investigation because both sides satisfy the same second order difference equation and the same initial conditions. The proof is now complete. 9 COROLLARY 2.2. The sn s have the generating function (2.5) 2 sn(x)f = 7^!!/' 2 Rn(Tk(x))fK n=0 1 — 2xt + t n = 0 Proof We have Downloaded from https://www.cambridge.org/core.