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They begin with a Proceflion, which commenc'd at the Porch, and ended at the Rail near the Al tar. His Homilies of the tenth Century, Vellum. We have nothing, but what is generally known, to fay con- cerning St. However, the publisher has no responsibility for the websites and can make no guarantee that a site will remain live or that the content is or will remain appropriate. He is the moft renowned Author of the Inftitutions and Collations, aid to have been Abbot of 5. Laerebog i matematisk Analyse, Vo! First note that by 7.

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Engage Promote your app as a way for visitors to engage your interpretive content through their own mobile devices. How can apps built with OnCell benefit your site? Basic hypergeometric series 30 Show that this function also satisfies the functional equation 1. Derive the generating function n f: This is a q-analogue of the formal differential equation for generalized hypergeometric series given, e.

Also see Jackson [Od, 15 ]. Exercises 33 Andrews and Askey [ ] 1. Andrews and Askey [] 1. Some techniques for using symbolic computer algebraic systems such as Mathematica, Maple, and Macsyma to derive formulas containing hypergeometric and basic hypergeometric series are discussed in Gasper [].

Bijective proofs of the q-binomial theorem, Heine's 2 Notes 35 sum in the form of a generalized q-binomial Vandermonde convolution. The even and odd parts of the infinite series on the right side of 1. Also see Suslov [] and the q-convolutions in Carnovale [], Carnovale and Koornwinder [], and Rogov []. Concerning theta functions, see Adiga et al. AI-Salam and Ismail [] evaluated a q-beta integral on the unit circle and found corresponding systems of biorthogonal rational functions.

It arises in such diverse fields as analysis, computer programming, geometry, number theory, physics, and statistics. A constructive proof was recently given by O'Hara []. Also see Bressoud [] and Zeilberger [a,b, b]. Also see Carlitz [a], Kadell [a], and Knuth [, p. Gasper derived the q-multinomial theorem in part ii several years ago by using the q-binomial theorem and mathematical induction.

Andrews observed in a letter that it can also be derived by using the expansion formula for the q-Lauricella function Notes 37 Ex. Pade approximants for the moment generating function for the little q-Jacobi polynomials are employed in Andrews, Goulden and D. Jackson [] to explain and extend Shank's method for accelerating the convergence of sequences.

Pade approximations for some q-hypergeometric functions are considered in Ismail, Perline and Wimp []. Kummer's summation formula 1. If the series 2. Dougall's [] summation formulas F. Note that Dixon's formula 2. Analogous to the hypergeometric case, we shall call the basic hypergeometric series '" [a l ,a2, It follows from 2.

Clearly, every VWP-balanced basic hypergeometric series is WP-balanced and, by the above observations, every WP-balanced basic hypergeometric series can be rewritten to be a VWP-balanced series of the form in 2. Then, by the q-Saalschiitz formula 1. This is equivalent to Bailey's [] lemma.

The most important property of 2. In the next two sections, like climbing the steps of a ladder, we will use 2. This formula follows directly from 2. Notice that the very-well-poised series in 2. Summation, transformation, and expansion formulas 44 but qa! For an early history of these identities see Hardy [, pp. It is clear that appropriate choices of A and Ak will lead to transformation formulas for basic hypergeometric series which have at least a partial wellpoised structure.

Note also that a terminating nearly-poised series of the second kind can be expressed as a multiple of a nearly-poised series of the first kind by simply reversing the series. By proceeding as in the proof of 2. To derive this formula, first observe that by 2. Summing the above 8 W 7 series by means of 2. It is sometimes helpful to rewrite 2.

On the other hand, if we take the limit n ; 00, we get the transformation formula for a nonterminating 8 8 1 2. It is a q-analogue of Whipple's [b] formula F 4 3 [-n, d,a, b, e, e. For example, if b, c, or din 2. However, the terms near both ends of the series on the right side of 2.

The procedure is schematized as follows: Letting m follows from 2. A special case of 2. However, we can still express 2. First, use Heine's transformation formula 1. However, the conditions under which 2. The notational compactness of 2. In addition, the symmetry of the parameters in the q-integral on the left side of 2. The advantage of our use of the q-integral notation can be seen by comparing the above proof with that given in Bailey [].

We now turn to the double sum that corresponds to the lower limit, A, in the q-integral 2. Use of this breaks up the double series in 2. Equating the expression in 2. Bailey [], Carlitz [a] 2. This formula is equivalent to Jain's [, 4. Sears [c,d], Bailey [] ii Deduce that q!

Ismail and Rahman [b] 2. See Jain and Verma [] and Gasper and Rahman []. Note that this reduces to 2. Note that this is a nonterminating extension of 2. These formulas are q-integral analogues of Erdelyi's [, equations 17 , 11 , and 20 , respectively] fractional integral representations for 2Fl series. Additional constant term results are derived in Baker and Forrester [, ], Bressoud [], Bressoud and Goulden [], Cherednik [], Cooper [a,b], Evans, Ismail and Stanton []' Forrester [], Kadell [, , ], Kaneko [], Macdonald [b], Morris []' Opdam [], Stanton [b, ], Stembridge [], and Zeilberger [].

The main topic of this chapter, however, will be the q-analogues of a large class of transformations known as quadratic transformations. So we shall just say that a transformation between basic hypergeometric series is "quadratic" if it is a q-analogue of a quadratic transformation for hypergeometric series. It will be seen that one important feature of the quadratic transformations derived for basic hypergeometric series in the following sections is that the series obtained from an r 3F2 3.

Let us consider this class of formulas first. Case iii Let c t 00 in 3. This gives Sears' [c, 4. Case iv Replacing a by aqn in 3. Case v In 3. Before leaving this section it is worth mentioning that by taking the limit n ; 00 in Watson's formula 2. As a special case of 3. We start by replacing a, b, e, d, e, j in 2. Some of these transformation formulas also arise as special cases of the more general formulas that we shall obtain in the next chapter by using contour integrals.

An important application of 3. If we now replace d by dqn in 2. In terms of q-integrals formulas 3. Similarly we can express the second double series on the right side of 3. Combining the two we get 3. The special case of 3. Notice that the part of the series on the left side of 3. To derive a useful extension of 3. It is these observations and the factorization that occurred in 3.

If we replace n, a, band k in 3. Since inverse matrices commute, by computing the jkth term of BA, we obtain the orthogonality relation t. For applications to q-analogues of Lagrange inversion, see Gessel and Stanton [, ] and Gasper [a]. Special cases of 3. Verma [] showed that 3. Then by multiplying both sides of 3.

For basic hypergeometric series, 3. Note that by letting er t 00 in 3. Other types of expansions are given in Fields and Ismail []. AI-Salam and Verma [] used the b t 0 limit case of the summation formula 3. In order to employ 3. By replacing An, B n , x, w by suitable multiples, we may change 3. Multivariable expansions, which are really special cases of 3.

For a multivariable special case ofthe AI-Salam and Verma expansion 3. Thus, 2 2 2 2 bf l - c l - aclb g ac Ib;c,d,e,f,aclb,cqlb,cq Ib;q ,q - ac2 1-! More generally, we shall call a series Similarly a bilateral series Multibasic series are sometimes called polybasic series. Therefore, in working with multibasic series either the series are displayed explicitly or notations are employed which apply only to the series under consideration.

Similarly, use of the expansion formula 2. By reversing the series on both sides of 3. Thus, we obtain the summation formulas: Since the series on the left sides of 3. Now that we have the summation formulas 3. Show that this formula is a q-analogue of Gauss' quadratic transformation formula 3.

Koornwinder in a letter suggested part iii. Evaluate the sum when n is even. Nassrallah and Rahman [] 3. Gasper [a] Exercises 3. O show that 3. Rahman [] 3 3] q-n-l, q-n, ql-n, bcdq3n b d ; q ,q q,cq, q Exercises 3. Gasper and Rahman [] 3. Suslov [, ] 3. Inversion formulas are also considered in Carlitz [] and W.

Chu [b, ] and, in connection with the Bailey lattice, in A. Agarwal, Andrews and Bressoud []. Jackson [, ] and Jain [a] also derived q-analogues of some of the double hypergeometric function expansions in Burchnall and Chaundy [, ]. Krattenthaler [b] independently derived the terminating case of 3.

For further results on cubic and quintic summation and transformation formulas, see Rahman [d, b, ]. Other q-analogues are given in N. Agarwal [], and Jain and Verma []. Applications of the Fundamental Lemma to mock theta functions and partitions are contained in Andrews [a,b].

Agarwal [a] extended Andrews' Fundamental Lemma and pointed out some expansion formulas that follow from his extension. The assumption that there exists such a contour excludes the possibility that a or b is zero or a negative integer. Barnes' first and second lemmas are integral analogues of Gauss' 2Fl summation formula 1. In Askey and Roy [] it was pointed out that Barnes' first lemma is an extension of the beta integral 1.

Following Watson []' we shall give a q-analogue of 4. We shall give q-analogues of 4. The rest of the chapter will be devoted to generalizations of these integral representations, other types of basic contour integrals, and to the use of these integrals to derive general transformation formulas for basic hypergeometric series.

Re[slog -z -log sin1rs ] Basic contour integrals from the poles as shown in Fig. It should be noted that the contour of integration in 4. To see that 4. Barnes [] used 4. Since the residue of the integrand at A q-analogue of Barnes' second lemma 4. Next, replace c by d in 4.

Then the series on the left of 4. The symbol "idem ali a2, Thus, it follows from 4. This gives Bailey's transformation formula 2. Evaluating the above integral via 4. The integral in 4. However, here we shall let m be an arbitrary integer. Special cases of 4.

Although each of the above three types of integrals can be used to derive transformation formulas for basic hypergeometric series, we shall prefer to Basic contour integrals use mainly the contour integrals of the type in 4. This is formula 5. Observe that by replacing 4. In Askey and Roy [] it is also shown how Barnes' beta integral 4.

Analogously, application of the summation formula 2. In addition, application of Bailey's summation formula 2. Slater [c] observed that this formula could also be derived from 4. Show that the q-Cauchy beta integral in Ex. If s 1, then the first series converges for Izl R. If we replace the index of summation n in 5.

The very-well-poised bilateral basic hypergeometric series in 5. Note 1 that if in 5. The well-poised bilateral basic hypergeometric series in 5. Andrews [, a], Hahn [b]' M. Jackson [b], Ismail [] and Andrews and Askey [] published different proofs of 5. The proof given here is due to Andrews and Askey [].

Jacobi's triple-product identity 1. First replace a and z in 5. Slater and Lakin [] gave a proof of 5. Andrews [a] gave a simpler proof and Askey [c] showed that it can be obtained from a simple difference equation. First observe that 5. As in Slater [b], Sears' formulas 4. Replacing a, b, e, d, e, j, 9 by a 2, ba, ea, da, ea, j a, ga, respectively, we may rewrite 5.

See the Notes for this exercise. Bailey [a] and Garrett, Ismail and Stanton [] 5. See Askey and Wilson [, pp. Formal Laurent series and Ramanujan's sum are considered in Askey []. A probabilistic proof of 5. Milne [, a, ] derived multidimensional U n generalizations of 5. Jackson [] employed 5.

Some recent results on bilateral basic hypergeometric series are given in Schlosser [a,b,c]. For various proofs of the quintuple product identity and of its equivalent forms and applications to number theory, Lie algebras, etc. Notes For integrals of Ramanujan-type that correExercises 5.

A significant extension of the beta integral was found by Askey and Wilson []. Since it has five degrees of freedom, four free parameters and the parameter q from basic hypergeometric functions, it has enough flexibility to be useful in many situations.

This integral is j 1 1 1 h x;I,-I,q"2,-q"2 dx -1 h x; a, b, e, d viI - x 2 2n abed; q oo q, ab, ae, ad, be, bd, ed; q oo' 6. Askey and Wilson deduced 6. Askey and Wilson's original proof of 6. In their paper 6. Simpler proofs of 6. In the following section we shall give Rahman's proof since it only uses formulas that we have already proved, whereas the Ismail and Stanton proof uses some results for certain orthogonal polynomials which will not be covered until Chapter 7.

For the integrand in 6. It is the freedom provided by these extra parameters which will enable us to prove a number of important results in this and the subsequent chapters. It is easy to check that, by 2. By analytic continuation, the restrictions on a, b, e, d mentioned above may be removed.

To derive it, replace a,b,e,d,f,g in 6. Also, replace x in the integral 6. Replace j by jqn in 6. Since the integrand on the left side of 6. This provides an alternate proof of Bailey's four-term transformation formula 2. By applying the iteration of the transformation formula 2. Similarly, by applying 2. Using this in 6. This gives a nonterminating extension of the transformation formula 3.

For an extension of 3. Splitting h x; b in the same way as in 6. The reduction of this 8 W 7 will be done in two stages. First we use 2. It is now clear that we can handle the cases of three or all four of the parameters a, b, c, d exceeding 1 in absolute value in exactly the same way. If m side is to be interpreted as zero.

Prove that if j and k are nonnegative integers, then C. See Gustafson [b] and Rahman [a] 6. Aomoto [] considered a generalization of Selberg's integral and utilized the extra freedom that he had in his integral to give a short elegant proof of it. Habsieger [] and Kadell [b] proved a q-analogue of Selberg's integral that was conjectured in Askey [b].

For conjectured multivariable extensions of the integrals in Exercises 6. Also see the extension of Ex. A finite or infinite sequence Po x , Pl x , This characterization theorem of orthogonal polynomials is usually attributed to Favard [], but it appeared earlier in published works of Perron []' Wintner [] and Stone []. For a detailed discussion of this theorem see, for example, Atkinson [], Chihara [], Freud [] and Szego [].

In the finite discrete case the recurrence relation 7. In general, the measure in 7. However, for a wide class of discrete orthogonal polynomials it is possible to use the recurrence relation 7. We shall illustrate this in the next section by considering the q-Racah polynomials Askey and Wilson [].

This enables us to rewrite 7. A straightforward calculation gives 7. The verification that 7. N is left as an exercise Ex. The connection between the 6j-symbols and the 4F3 polynomials 7. The q-analogues in 7. We shall now point out some important special cases of the q-Racah polynomials. For some applications of q-Hahn, dual q-Hahn polynomials, and their limit cases, see Delsarte [a,b, ], Delsarte and Goethals [], Dunkl [] and Stanton [c].

Then the orthogonality relation 7. It is easy to verify that Pn x; a, b; q satisfies the three-term recurrence relation 7. In view of the third free parameter in 7. In his work of the 's, in which he discovered the now-famous RogersRamanujan identities, Rogers [b, , ] introduced a set of orthogonal polynomials that are representable in terms of basic hypergeometric series and have the ultraspherical polynomials 7.

For further results on the asymptotics of Cn x; ;3lq , see Askey and Ismail [] and Rahman and Verma [a]. Since the integrand in 7. From Sears' transformation formula 2. In addition, for real 0 these polynomials are analytic functions of a, b, c, d and are, in view of the coefficient ab, ac, ad; q na-n, realvalued when a, b, c, d are real or, if complex, occur in conjugate pairs.

Askey and Wilson [] introduced these polynomials as q-analogues of the 4F3 polynomials of Wilson [, ]. Since they derived the orthogonality relation, three-term recurrence relation, difference equation and other properties of Pn x; a, b, c, dlq , these polynomials are now called the AskeyWilson polynomials. Since the three-term recurrence relation 7. So by Favard's theorem, there exists a measure da x with respect to which Pn x; a, b, e, dlq are orthogonal.

First observe that, by 7. Askey and Wilson proved a more general orthogonality relation by using contour integration. For a proof and complete discussion, see Askey and Wilson []. The proof of 7. Following Askey and Wilson [] we shall obtain what are now called the continuous q-Hahn polynomials.

First note that the orthogonality relation 7. Pm cose; a, b, c, dlq Pn cose; a, b, c, dlq w cose; a, b, c, dlq sine de hn a, b, c, dlq ' 7. If the polynomials fn x happen to be orthogonal on an interval I with respect to a measure da x , then Ck,n is the k-th Fourier coefficient of gn x with respect to the orthogonal polynomials fk X and hence can be expressed as a multiple of the integral II fk X gn x da x.

A particularly interesting problem is to determine the conditions under which the connection coefficients are nonnegative for particular systems of orthogonal polynomials. See the applications to positive definite functions, isometric embeddings of metric spaces, and inequalities in Askey [, ], Askey and Gasper []' Gangolli [] and Gasper [a].

The 5rP4 series in 7. One of the simplest cases is when the 5rP4 series reduces to a 3rP2 , which can be summed by the q-Saalschiitz formula. It is left as an exercise Ex. For other applications of the connection coefficient formula 7. Generally, for a differentiable function.

X Applications to orthogonal polynomials Following Askey and Wilson [], we shall now use the operator Dq and the recurrence relation 7. First note that by 7. Hence show that 7. Askey and Wilson [] 7. Ismail and Zhang [] 7. Ai-Salam, Allaway and Askey [b] Exercises 7.

AI-Salam, Allaway and Askey [b] 7. Gasper and Rahman [] 7. Le iIJ , aJ. Le- iIJ , bJ. Le- iIJ ; q oo abJ. L- 1 Iq , where max lal, Ibl, IJ. Gasper [b] Exercises 7. See Ai-Salam and Garlitz [, ], Ghihara [, Deduce that the polynomials hn x; q , which are called the Rogers-Szego polynomials, satisfy the Thnin-type inequality for x ; Note that this defines a formal inverse of the Askey-Wilson operator D q.

Ismail and Rahman [a,b] 7. For a classical polynomial system with complex weight function see Ismail, Masson and Rahman []. The familiar connection between continued fractions and orthogonality was extended by Ismail and Masson [] to what they call R-fractions of type I and II, which lead to biorthogonal rational functions.

See further work on continued fractions related to elliptic functions in Ismail and Masson [], Ismail, Valent and Yo on [], and Milne []. Chihara and Ismail [] extremal measures for a system of orthogonal polynomials in an indeterminate moment problem are examined. For orthogonal polynomials on the unit circle see Ismail and Ruedemann []. Some classical orthogonal polynomials that can be represented by moments are discussed in Ismail and Stanton [, ].

Koelink and Koornwinder [] showed that the q-Hahn and dual q-Hahn polynomials admit a quantum group theoretic interpretation, analogous to an interpretation of dual Hahn polynomials in terms of Clebsch-Gordan coefficients for SU 2. For how Clebsch-Gordan coefficients arise in quantum mechanics, see Biedenharn and Louck [a,b]. Chihara [] considered the locations of zeros of q-Racah polynomials and employed her results to prove non-existence of perfect codes and tight designs in the classical association schemes.

The correspondence between q-Racah polynomials and Leonard pairs is outlined in Terwilliger []. For the relationship between orthogonal polynomials and association schemes, see Bannai and Ito [], L. Chihara and Stanton [], Delsarte [b]' and Leonard []. A multivariable extension of the q-Racah polynomials is considered in Gasper and Rahman [c], while a system of multivariable biorthogonal polynomials is given in Gasper and Rahman [a], which are q-analogues of those found in Tratnik [b] and [], respectively.

In Ai-Salam and Ismail [] they considered a related family of orthogonal polynomials associated with the Rogers-Ramanujan continued fraction. A biorthogonal extension of the little q-Jacobi polynomials is studied in Ai-Salam and Verma [a]. Since the Hamburger and Stieltjes moment problems corresponding to these polynomials are both indeterminate, there are infinitely many nonequivalent measures on [0, 0 for which these polynomials are orthogonal.

Chihara [], and Shohat and Tamarkin []. Ai-Salam, Allaway and Askey [a] gave a characterization of the continuous q-ultraspherical polynomials as orthogonal polynomial solutions of certain integral equations. Askey [b] showed that the polynomials Cn ix; jJlq , Ismail and Rahman [] showed that the associated Askey-Wilson polynomials r;:: A survey of classical associated orthogonal polynomials is in Rahman [], and an integral representation is given in Rahman [b].

A projection formula and a reproducing kernel for r;:: Berg and Ismail [] showed how to generate one to four parameter orthogonal polynomials in the Askey-Wilson family by starting from the continuous q-Hermite polynomials. Also see Koelink [b] and Rahman and Verma []. Kalnins and Miller [] employed symmetry techniques to give an elementary proof of the orthogonality relation for the Askey-Wilson polynomials.

Following Hahn's approach to the classification of classical orthogonal polynomials N. Atakishiyev and Suslov [b] gave a generalized moment representation for the Askey-Wilson polynomials. Brown, Evans and Ismail [] showed that the Askey-Wilson polynomials are solutions of a Notes q-Sturm-Liouville problem and gave an operator theoretic description of the Askey-Wilson operator V q.

Chihara [] extended her work on q-Racah polynomials [] to the Askey-Wilson polynomials. Floreanini, LeTourneux and Vinet [] also employed symmetry techniques to study systems of continuous q-orthogonal polynomials. A multivariable extension of Askey-Wilson polynomials is given in Gasper and Rahman [b] as a q-analogue of Tratnik [a], see Ex.

For more results on contiguous relations and orthogonal polynomials, see Gupta, Ismail and Masson [, ], Gupta and Masson [] and Ismail and Libis []. An operator calculus for Dq is developed in Ismail [a]. In Koornwinder [] it is shown that the orthogonality relation for the q-Krawtchouk polynomials Kn x; a, Nlq expresses the fact that the matrix representations of the quantum group Sp,U 2 are unitary.

Chihara and Stanton [] showed that the zeros of the affine q- Krawtchouk polynomials are never zero at integral values of x, and they gave some interlacing theorems for the zeros of q- Krawtchouk polynomials. Atakishiyev and Klimyk [] found the connection between big q-Laguerre and q-Meixner polynomials and representations of the group Uq SUl,l.

Ciccoli, Koelink and Koornwinder [] extended Moak's q-Laguerre polynomials to an orthogonal system for a doubly infinite Jacobi matrix originating from analysis on SUq l, 1 , and found the orthogonality and dual orthogonality relations for q-Bessel functions originating in Eq 2. For derivations of the addition formula for Jacobi polynomials, see Koornwinder [a,b] and Laine [].

In view of the two different orthogonality relations for the q-Laguerre polynomials, it follows that there are infinitely many measures for which these polynomials are orthogonal. The Stieltjes- Wigert polynomials see Chihara [, pp. Askey [] gave the orthogonality relation for these polynomials with a slightly different definition that follows as a limit case of the first orthogonality relation in this exercise.

Ai-Salam and Verma [b,c] studied a pair of biorthogonal sets of polynomials, called the q-Konhauser polynomials, which were suggested by the q-Laguerre polynomials. For asymptotics of basic Bessel functions and q-Laguerre polynomials, see Chen, Ismail and Muttalib []. This formula does not extend to noninteger values of n because, in general, the triple sum on the right side does not converge.

Thus, we obtain the formula bb3c2jaq, q where. Application of Sears' transformation formula 2. Two particularly interesting ones are Further applications 8. Either of the formulas 8. The special case in which the lOcP9 series in 8. This provides a q-analogue of Bateman's [, p. This is a Watson-type formula.

Two additional Watson-type formulas are given in Ex. Letting c t 0 in 8. In fact, if we replace a and c by q 2o: For terminating series there is really no difference between the Watson formula 8. However, for the continuous q-ultraspherical polynomials given in 7. Sixtus iiVftspq tbpJR. Of them Spartian fpeak3 thji. Sabiua ; the Infer ipt ions there ; the.

This he borrow'd out of Oviddefafiitj lih. Here was. The Church is fupported on both fides with Fluted Columns. Among them is the R. Dontwick's Head, but mift his Aim. Clofe by is the Church of St. MextuJfy of the Je- ronymites, of which we have nothing new to fay, tho' it. Adjoyning to it is the Priory oiSt.

On the other fide of the Hill is the Church of St: The Body of St. Perer baptized Prifca. On the fame Aventine Hill is the Church of St. Sa- -iasj which wasalfo one of the twenty four Abbeys, handsome enough, and now belonginjg to a College of JefuitF. This Sta-. This Cell had neither Door nor Window,.

Sabas;, in the Vineyard now belpnging to -D. I never tourid that Rabbits were us'A in Lupercd Worlhip. Lyon devouring t No fe. The Ifle is fupportcd ;hty fluted Columns. In the Choir thete are two nns of Egyptian fpeckled Marble, iiot inftrior in and Height to the Columns of the Pantheon and dory de jingelis. There are many more rulars concerning this Church in the Defcriptions we, and it belougsto our Monks oi Monte daffmo.

Martina dni St. Yet there are fome who en- deavour to place both the Fortrefsand Temple of? Nothinjg now occurring that may incline us to cither fide, we refer this Matter to the more SkiL fuL This Temple was two hundred Feet in Lengthy and wanted but fifteen of the fame Breadth. The Roman Knights who had flood very thick on the Steps of Concord. The other Things relating to the Arf h are every where to be found in Print.

From the Chappel they go down into the deep Dungeon, where is the Spring miraculoully produced by St. Over the Mamerttne Goale is the Church of St. Jo- fephj aflign'd to the Carpenters. Some fay the tui- tion Prifons were at St. Clofe by, on the other fide of the Lane fjands the Church of St.

Martina or of St. In the fame Place is the Church of 3f. The Temple here built by the Emperor M. The Capitals are adorn'd with Monfters and Foliage. Almoft adjoyning to Faufllna's Temple is the Church of St. But perhaps it hap- pened by this as Suetonius tells us of. Days of Pope Adrian I. Marcus and Mar- cellianus and others, as alfo of Felix II. Thefe are the Farnefian Tables frequently mention d in this Di- ary.

Hi' ' ther was brought all the Treafure and Precious Vef- fels of the Temple of Jerufalem, and it was lin'd throughout on the infide with gilt Brafs Plates. The Portico of this Temple, which confifted of fix Columns. Ih that fame Tem- ple was the Library Aulus Gellius fipeaks of j nor ought we to omit making mention of the Statue of the River Nile fpoken of by Pliny lib. Mary Nova or St.

Knees is ftill to be feen. There is alfo the Pifture of the Blefled Virgin, which they fay was painted by St. Luke and brought to Rome from Greece. FLA- 3o6 F. For fince it is apparent that the Golden Houfe ftood in this Place, which the An- cient fay was of a ft upend ious Magnitude and Magnifi- cence, it may be proper to aflign fo noble a Pavement to it.

What thoie Arches, which are ftill ftanding in the Orchard of St. The Farnefian Gardens, which take up all the top of the Palatine Hill, for the moft part formerly be- longed to the Imperial Palace. The Name of a Palace was deriv'd from the Palatint Hill. At the firft En- trance -loS. We enquired of the Gardiner, where that Marble Table was. He anfwer'd, all had been car- ryd away, and nothing remain'd.

All the Hill is full of fubterraneous Paflages, the Entrance into which was purpofely ftopt up. On the other fide of the Hill towards St. King Tbeodoricus difcourfes notably concerning the.. Neither muft it he though. THofeRitoWeteftolnaway, only fome icw rtiniaining, tha! NardhTvs gutk'd tin t: There qrealfbmany HouSs.

Mary the Deliverer, where was fomd an admirable Ba s Relieve of M. Prpius iter noftris oftendit in aethera divisi That is. Terga Pater, blandoque videt Concordia yultu. We here iir; ierttheDraughtofjt. On it ard curioufly carv'd the Infl: Hard by is a fmall Stream of Water, which having been difcover'd of late Years, broke out, and runs into the greateit,aacient common Sewer.

The Common Sewers are reckoned among the Wonders of J? Denis converted into the Church pf St. Mary in Cofincdin. It appears by ancient Tradition that St. Impijs fuperftitionibus Deoram. In Englijh. It is now caird the Church of St. Oppofite to it is a ChappelcaUM St. They begin with a Proceflion, which commenc'd at the Porch, and ended at the Rail near the Al tar. With thefe In- ftruments they made a Noife at the Bifhop's Ears,, and they founded the louder at certain Parts of the Litur- gy.

The Greeks call this fort of Inftrunicnts Anarri- pidia. Tie Seventh Daj. The outward Front of the Church is adorn'd with a curious Portico. Clofe by is alfo the Monaftery of 5it. Here is the Pifture of Bartolusy the Civilian, by Raphael. J Kot far from bence is the "Monaftery.

Thatineu The Clmrch is curioufly aclorn'd. TbeV vvsre brought to. Rame by Confiamiue ttie Great, an3 rlacd atjiis B-uhs. On the Pedeftal of them is this ir. But Fhidhn and Praxiteles having liv'd long htiote Als. Flaminius peaks quite. On thole fame Pedeftals flood two Marble Cafiors, carry'd into the Capitol by? I believe this is the Statue that is in the.

Sylvefter, difcover'd foiiie broken Arches, full of Earth fallen in, and beoan to empty thole Vaults, where he found many Pieces of Columns ' of the Marble us'd for Statues, thirty Spans long, ' with fome Capitals and Bales. Being my Friend, he fent for. Atlength we proceed- j ed fo far as to come to the End of tJie Vaults. Mary Major is the: My PredeceflToj?

They there found Hercules's Head by a skil- Hand, but worn away. In it is the Church of St. Whit Country Man are yoa? Do you. S tQ how. The Idol held both Hands on its Breaft, and a Key in each of them. Before I deliver my Opinion concerning this Matter, we will hear our noble Carver peaking of an Image very like this.

Wp here give tlie Draughts of tlieip both. The fame is alio plainly deliver'd by Tertulitan in his Apology, and St. Mithras is not always drawn alike, nor does he of- ten occur with a Lion's Face among Hafs Relieves and. This vy puld have been acceptable to that learned Man, who had heard notliing ofany fuch Fi- gure of Mithra.

The Serpent is always drawn with him in fe- veral Figures, and I never yet fiw any without it. Fiamrn'tus alfo mentions a Statue wound about with a Serpent, where he Ipeaks above of the Church of 5Sr. Macrohius lih. But who can difcover why different Numbers of Wings are afiigny, fometimes two and fomctimes fotfr?

Laftly Mithras has a Globe under his Feet, either becaufe. Greek Manufcript of Gregory Nazjanxjen. We find thus much in it concerning Mithras, The forty feventh Sto- ry exprejfes the Torments of Mithras. This is un-: I k it down that fame Year, on the 1 7th of March, as Hows. But it appears by many Inftances,thit.

Pudentiana caft the Bodies of three thoufand Martyrs. The Church of iSr. Mary in AfontihnSj or. Thus the Church of St. Saviour in VeBurey might have beenfocall'd, becaufe Handing in: The Monaftery Flam: Dominickj inftituted for inftrufting of new Converted Women. Nervals Forum wajotherwife call'd TranJitoriuMj that is, the Paflage Market Place, becaufe being feated in the mdl frequented Part of the City, there was always a great Concourfe of People palling to and fro.

C HAP. The Eighth Daj. Agnes and St. Bernard of the Fulienfes add much to the Beautifai Profpeft. But thofe which aretalL t,heing fet deep into the Ground areoii- ly bmuch above it as to be equal with the reft. Among thefe Ruins of the Baths, and in the Rooms adjoyning, there are Remains of Marble cafing, which the Carthufians have made u e of to adorn their Church anew, Charles Maratta' being their Arcbiteft.

The -loufc adjoyning is fet off with many Ornaments,and in t is a Bafs Relieve of King Pyrrhus by forae great Mai- ler. Clofe by is a Hill enclos-4 Vxth Cyprefs Trees, with Rome as big as a Colojfus ' ittingon it, which I fuppofe is the fime mention d by Hamlnhis in the fourteenth Cliapter.

Thus much of thv fe noble Gardens, to defcribe the whichi would require a large Volume. The reft of it is generally known to moft Men. The other Things, s Bacchus s Tomb, the ancient PiAures and the like, lave been very often fpoken of. Behind St. Cliance firwardedthe. I have been told fach n Image of St.

Returning the fame Way we go to the Barberine Palace, than which none is more magnificent through- out the City. Among thofe that remain I took Notice of thefe that follow. I am of Opi- nion there is no other Copy of that Liturgy fo anci- ent: Clergy, fo tlwt they perform'd the Divine Service m.

Years fince, being tir'd with the frequent Refgrt oi. Contacium is a very fhort Staff, to which is made feft and wrapp'd round a flip of Parchment of a vaft Length, confifting of many Pieces pafted together -, on which are written the Prayers, and Offices to be performed by the Priefts at the Di- vijic' Service. This is a noble Addition to other Colieftions out of other Works, which we are making ready for Publication.

It is a Manufcript of the eleventh Century, curioufly writ on Vellum. The Books of the Prophets, Vellum, of the eleventh Century. Danhl the Prophet, and fome Homilies, Vellum, of the eleventh Century. Two Manufcripts of the Gofpel? We fcgkei of the Colojfean Head 'above at the Filla Mai- theia. In the midft of tM larble Stone, between the Emperor and the Em- prcfs, rifes as it were a Table for fome Infcriptibn, which does not appear.

It ik here dcfcrib'd for its fingularity. The upper Pavement " was of Marble of feveral Colours curioufly laid. PfW V in Momorio were adorn'd. Aurelius Pacoras, and M. That fubterraneoas Part fo adorn'd with Pipes, is fuppos'd to have been built for the Priefts to walh before they perform'd the Religious Rites.

Ftdvius tells us he aw the Qones of one of them found in that Place. Garden Hill. It would be tedious to mention them all. VlXir ANN. Nicholas de Pel- Icve, Cardinal of St: Antony Lanfrancus? The Church is notably adorned with Paintingsi but the nobleft is the Piece which reprefents the ta- king of our Saviour clown from the Croft, the like whereof is fcarce to be found in Rome in Frefca Pope Sixtus V had b great a Value for it, as to order it to be carry'd with that Piece of the Wail to the Vatican ; but the Monks reprefenting how difficult it would be to remote fuch a MaC without en- dangering the Wall, and fpoiling the Painting, he forbore.

He was the moft renowned Philofo' Eher of the feventeenth Century. There is nopleafanter Place in aU the City: V Jonrptey comes the Seat of fo great a Prince. The Gardens are enclosed with the City Wall, on the fide next the Fields At the Entrance into tlie Garden are thofe two Lions, only the half of one of which is ancient, for having been formerly a Bafs Relieve, and the Thicknefs of the Marble bfeing fufficient to makeaneifc' tire Lion, the whole true Proportion was made up by John Scerano.

Flaminius Vacca has more than once told us above, that he carv'd fuch another Lion, that there might be one oppofite to the other for Unifor- mity fake. Here is al b that Silfnus holding the Infant Bacchus in bis Arms, which was faid above to have been dug up in the So- iuftidn Gardens. In the Garden ftands a fmali Strufture, pro. From thefe two Significations they divide into feveral Qpi- nions, I am more inclinable to the latter.

On one ficjc of the Street aopears the Church of St. Our noble Canrec gives this Accouht df thein. Mary in Fla. Charles of the Mila-? Charles, in tlie Houfe of Fiortrvante are Remains of an. The Thirteenth Day. We began the' Twelfth Day at the Church of St. Lanrence in Lucina. It was de- dicated to the Sun, as the Infcription declared.

The Virgin Water, as may be feen in Caffio- darns Form, 6. The Goth made fuch a ftrift " Search, that he found thofe Symbol? What Flaminius here relates, does not in the Icafl alter the Opinion! Goth muft have receiv'd that Symbol by Inheritance, delivered down from his Anceftors to his Age, after above eleven Centuries, which is not at all likely 9 or elfe he muft have found out the Place and Token of the Treafure, at fo great a diftance, by a Diabo- lical Art, which fcarce any Man in his Senfes will be- lieve.

Nothing is here more remarkable than the Library, which is inferiour to very few for the Num- ber and Excellency of printed Books. Befides it is well ftor'd wnth Greekjiud Latin Manufcripts. I took the following Note of the Greek. Another Manufcripts on Silk, containing all the Prophets, with fome Things ftruck out, and Afterifcs or Stars, and fome Hexaple Readings, that is of the fix feveral Tranflations, in the Margin.

Another modern Manufcript with many Readings of the Hexapla or fixfold Tranflation. There is alfothc Hebrew Bible moft curioufly written, and the Gofpek in Armeniack. Bafil of Baptifm. His Homilies of the tenth Century, Vellum. Gregory Nauanzjeifs Orations, of the eleventh Century, Vellum.

His Epiftles of the fourteenth Century Silk. His Poems, Uves and other things. The Lives ofthe Saints in three Volumes, Vellum,. PandeElsy or Colleftions out of feveral Fathers, of the fourteenth Century. Helios Cretenfis on St. Some Wor k s of Emanuel Chryfolora.

I ne- ver faw any Book of that Authors fo ancient and lb fine. Libanius's Orations and Declamations, of the tenth Century, Vellum. Proclus on the Tintaus. He never denies Accefs to any Man that is addifted to Literature, and keeps very learned Men in his Houfe. Saluft of the fame Age. Sdvian of the fame Age. On the Altar is a finall Pifture done by St. Under it is a Cave, or Vault, where they tell us St.

In a little wooden Box fome of St. Marliatjus confirms the fame Thing. There wdn. Face hiding that of Olympias. And M. In the faid Prince Livio's Mufeum there are many MarbleStatues and Images, leveral by very able Ma- ilers -, asalfo noble Paintings, extraordinary delightful for their Rarity and Beauty.

There are al b many Pieces of the moft famous Painters, remarkable for "I their Number and Value, which, according to our Cuftom we forbear to enumerate. Bafil, in which they had liv'd till then, for that of St. Lukcy which is held in ' great Veneration to this Day. Hiacinthus 4e NobiUj ,.

Hiftory of this Monaftery in Italian. Hard by is the little Hill now call'd Monte Cimi9. The Charter runs thus. It plainly appears by the Churches that the Hill there caird Mons Accept a- bilis is the fame now nam'd Monte Citorio. It was af- terwards caird Afons acceptcrittSj and at laft by corrup- tion Cttorio.

Having gone through a fmall Lane there is in. Maurus were levell'd with the Ground to make room for this Pile. In this Mona- ftery refides the General of the Order F. A ftupendious Work built by M. Now what Otcafion would there have been for bringing Marble fo far, if fuch mighty Stones could have been made at Rome?

For it is no wonder we fhould not know where that Quarry is, fince thofe Parts have beenalmoftin- acceffible to Chriftians for near eleven hundted Years ; all Egypt having been fubdu'd by the Mahometans in the feventh Century, and ever fince continued fub- jeft totnfidels of that fame fort, tho' tlie Sovereign be chang'd, thofe Quarries liave lain neglefted and for- gotten.

FLA- s86 F. The Architefts of the. The fame Day we faw the Palace oiStrozjLt. There the moft Illuftrious Leo Strozx. An entire Series of Gold Coins, among whichare many fcarceand very valuable Pieces. That moft courteous Gentleman gave us leave to tranfcribe this Manufcript.

The Fourteenth Day. The MonAJlery of St. I tranfcrib'd feveral Charters from the Archives of that Monaftery. Our Carver mentions fonie things dug up here. Aurelitu in Armour j and we Forbear to mention the reft for fear of doing what has 3cen done before. Mary Major. Nor is Na - dintis to be regarded, who rejefting this Natural E- tymology, has givei? Not far off, next the Church of 5f.

There are two Statues of St. The Eminent Cardinal G? We took a very large Account of tliefe Manulcripts, which is too long to find Place in this Diary. There is alfo great ftore of Coins of tlie large fize, and a curious Series of Em- perors. Adjoyning to this Houfe is the Church of St. On the Way to the Church of St.

Cardinal de Maxlmis had a Copy of them taken before that happened, as it is now preferv'd in that Palace. Petrus cum fratre Francifco Maximns, ambo Huic operi optatam comribuere Domum. We leave ihii; to be farther examin'd by the Learned, what has been faid gives occafion to enquire more into this Mattfer.

Near this Palace is the Church of St. Not far from hence is that Palace of Farnefij fo much celebrated throughout the World, both tor its curious Strufture and Magnificence. Here is that Hercules fo nobly carv'd, Commodus naked carrying a Boy, the Hiftory of Dirce with many Figures cut in one Stone, all ot them well known.

Palfing thefe things by, we only nacp- tion thofe Flaminius reckons up, wnich will be accepta- ble becaufe they arf new. We muft here take. The Sixteenth Day. The Seventeenth Day. Mzxy beyond the Tiber; the Epitaph of Quotvultdeus dug up ; an extraordinary Bajon. A winding Staircafe within it goes up to tU top, where formerly flood the Statue of Trajan and the Urn with his Afhes, which being remov'd, the Statue of St.

Peter was fet up in its Place. But the lowennoft Line of its Infcription being worn out, many have endeavour'd to make it out, and fo run into feveral Opinions. To declare why fo lofty a HM and Place has been raifed withfuch mighty Works. No Man ought to fuppofc that the Anonymou? Itiswhite, but full of Spots, and of no great Value. Any thing might be expelled from " Trajan s Magnificence.

Clofe by is the Palace of St. Ad- joyning to it is the moft ancient Church and the Street of 5r. On one e fmaller fides a Cup, on the other a Difli or f which Symbols are common on Tombs. The ph is thus. F 3 a Marble Urn ten Spans long and two ifl t, with the following Inlcription.

AN XXX. On another. Pr CCC. In this fame Place feveral mofl beautiful Marble Stones, of various fprightly Colours, were dug up. JgnattHi lately built is fo full of Embellifhment? We find the fame Bcmd- " ing continued in leverat ad jacent Wine-Cellars. The former lib. Agrippa made a Laconick Hot-houfe. Pliny lib. He had alfo fi: Having view'd the Church of St.

In the Palace of the SabeBik is the. Statue of C. In each Temple was a curious Statue of its proper Deity. Thus Plirty lib. In the Garden of the Francifcans of St. We have nothing, but what is generally known, to fay con- cerning St. At the Foot of the Hill is the moft ancient Church of 5r.

Adjoyning to the Church of St. Mary beyond the n'lber is the Monaflery of St. In one of the Chappels is a Well, into vhich they fay St. Calixtm Pope was caft. The Aurelian Way was for- merly fet thick with notable Tombs, which are now all taken away.

Not for off is the Church of St. FLAMl- 5i8 F. The Nimteenth and Twentieth Days. The City of Thebes in Egypt. The Walls oi Babylon. The Maufoleum. Tiie Pyramids. The Temple of Adrian at CyzJcus. On the fame WalTas we go to St. Here lives the R. Vf r fo often, tnuugh he view it over.

Thus do they vary, as is ufual, about the Original of the Word Vatican. Was not ahcienter than the City. Pter'i Church an Um of th'. Pter, near the Altar " caird yultvs S. Thbugti I have feen Manufcripts as , ancient as this, yet none To perfeft and. The Colbert: Thefe are all in the uncial Charafter without tlic- Ac. I have feen very many other Mami'fpnpt. The Afts of the Apoftles with Gqlden Accents.

In another Place we read Courtois. The Horfe's Accoutrements reprefcnted there in the Figures have no Stirrops. There is alfo another Manufcript Terence of the ninth Cen- tury, writ, by one Noodogarius, as may be read io i I took Notice. We detign to publiih them in their proper Places.

The Tiier is oppofitc to the Nile with the Symbols by which he is known. As for thofe things! Many judicious Perfons are of Opinion, as well as my felf, that this Work which has been attempted by b many requires ftill a more accurate and skilful. Fir ft he. Situation, arid the Bulk bf t!

There is another Method hitherto nftgleiled,. Befides there have been ieveral Perlbns, who fipent moft of their Days in writing Defcriptions of the City, whofe Labours now lye conceal'd. In like manner Libraries are to be fearched for the mall Works of Writers of the middle Ages, who either made DeFcriptions, or Diaries, or Hiftories of the City, as for Inftance, thofe of the jtnmtymms Aq- thors, very often by us made ufe of in this Di- ary.

Virr0,who built almoft. Library of St.

In gallery victoria paris picture uploaded

An entire Series of Gold Coins, among whichare many fcarceand very valuable Pieces. Rahman, Mizan. A fractional Leibniz q-formula, Pacific J. For how Clebsch-Gordan coefficients arise in quantum mechanics, see Biedenharn and Louck [a,b]. Analogous to the name q-shifted factorial for a; q n, we also call a; q, P n the q, P -shifted factorial in order to distinguish it from the a, T-shifted factorial defined in the next section. National Park Service.

Basic Hypergeometric Series, Second Edition (Encyclopedia of Mathematics and its Applications):

Generalized Frobenius partitions, Memoirs Amer. If the polynomials fn x happen to be orthogonal on an interval I with respect to a measure da xthen Ck,n is the k-th Fourier coefficient of gn x with respect to the orthogonal polynomials fk X and hence can be expressed as a multiple of the integral II fk X gn x da x. Ivanov and S. Using 2. Jackson's work on basic Appell series and the works of R. Since in this chapter we will be mainly concerned with deriving and applying q-integral representations of q-Appell functions, in many of the formulas it will be necessary to denote the parameters by powers of q.

Some polynomials related to theta functions, Annali di Matematica Pum ed Applicata 4 41, Analogous to the first edition, papers that have not been published by November of are referred to with the year , even though they might be published later.


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