Contiguous relations, continued fractions and orthogonality

We examine a special linear combination of balanced very-well-poised basic hypergeometric series that is known to satisfy a transformation. We call this and show that it satisfies certain three-term contiguous relations. From two of these contiguous relations for we obtain fifty-six pairwise linearly independent solutions to a three-term recurrence that generalizes the recurrence for Askey-Wilson polynomials. The associated continued fraction is evaluated using Pincherle's theorem. From this continued fraction we are able to derive a discrete system of biorthogonal rational functions. This ties together Wilson's results for rational biorthogonality, Watson's -analogue of Ramanujan's Entry 40 continued fraction, and a conjecture of Askey concerning the latter. Some new -series identities are also obtained. One is an important three-term transformation for 's which generalizes all the known two- and three-term transformations. Others are new and unexpected quadratic identities for these very-well-poised 's.

     

Some theta function identities related to the Rogers-Ramanujan continued fraction

In his first and second letters to Hardy, Ramanujan made several assertions about the Rogers-Ramanujan continued fraction . In order to prove some of these claims, G. N. Watson established two important theorems about that he found in Ramanujan's notebooks. In his lost notebook, after stating a version of the quintuple product identity, Ramanujan offers three theta function identities, two of which contain as special cases the celebrated two theorems of Ramanujan proved by Watson. Using addition formulas, the quintuple product identity, and a new general product formula for theta functions, we prove these three identities of Ramanujan from his lost notebooks.

     

Rapid prediction and optimization of concentration conditions for preparative fractions by solid-phase extraction

SPE is an effective tool for concentrating preparative fractions isolated from a complex sample. To guarantee high efficiency and recovery of concentration, the concentration conditions could be optimized by predicting the breakthrough volume (V(B)). In this study, a method of predicting V(B )of unknown compounds in preparative fractions at any isocratic mobile phase composition with the analytical retention parameters a and c is described. The a and c values and the relationship between half peak width (W(1/2)) and retention time of a model analyte were measured using the analytical elution mode on an SPE column, and the V(B )and retention volume (V(R)) predicted with the a and c values were validated with breakthrough experiments. However, it is impossible to measure the a and c values of multiple compounds in a complex system directly on an SPE column with a low number of theoretical plates. The correlation of the a and c values between the SPE and analytical columns was developed so that the analytical data could be transferred to the SPE column. With the calculated a and c values, we could optimize the concentration conditions on the basis of the predicted V(B )and the volume of the preparative fraction.     

Determination of volatile and non-volatile nickel and vanadium compounds in crude oil using electrothermal atomic absorption spectrometry after oil fractionation into saturates, aromatics, resins and asphaltenes

In recent work, it has been shown that electrothermal atomic absorption spectrometry (ET AAS) can be used to differentiate between volatile and non-volatile nickel and vanadium compounds in crude oil. In the present work, the distribution of these two groups of compounds over different fractions of crude oil was investigated. For this purpose two crude oil samples were separated in two steps: firstly, the asphaltenes were precipitated with n-heptane, and secondly, the maltenes were loaded on a silica column and eluted with solvents of increasing polarity. The four fractions of maltenes eluted from silica column were: F1, saturated and light aromatics; F2, polyaromatics; F3, resins; and F4, polar compounds. Fractions F1 and F2 were further investigated using gas chromatography, and all fractions were characterized by CHN analysis, confirming the increase of aromatics in the fractions 2, 3, 4 and asphaltenes. For the determination of Ni and V by ET AAS, oil-in-water emulsions were prepared. The speciation analysis was carried out measuring without chemical modifier (stable compounds) and with 20 μg palladium (total Ni and V) and the volatile fraction was calculated by difference. The limits of detection were 0.02 μg g⁻¹ and 0.06 μg g⁻¹, for Ni and V, respectively, based on an emulsion of 2 g of oil in 10 mL. The volatile species of Ni and V were associated with fractions F3 and F4, while only thermally stable Ni and V was precipitated in part together with the asphaltenes.     

A Khinchin Sequence

We prove that the geometric and harmonic means of the sequence of positive integers proposed by Bailey, Borwein, and Crandall converge to the corresponding Khinchin Constants.

     

On the convergence of continued fractions at Runckel’s points

We consider the limit periodic continued fractions of Stieltjes type

Analytical performance of a multi-element ICP-AES method for Cd, Co, Cr, Cu, Mn, Ni and Pb determination in blood fraction samples

An analytical method using inductively coupled plasma atomic emission spectrometry (ICP-AES) for rapid simultaneous determination of seven heavy metals (Cd, Co, Cr, Cu, Mn, Ni and Pb) in human blood fractions, like plasma, cells fraction and whole blood, is performed. The optimum wavelength was selected using as criterions the sensitivity, the linearity and recovery of aqueous standard solutions. The pretreatment of the sample, the centrifugal conditions, the necessity of digestion and the dilution of the digests were also studied. For plasma it was possible to avoid digestion of the sample, but for cells fractions and whole blood the digestion is necessary. The samples were acid-digested by HNO₃ in closed Teflon tubes under high temperature and pressure conditions and were diluted before injection into ICP-AES. Also, optimization of the inductively coupled plasma conditions like nebulizer argon flow rate, sample flow rate and power of radio frequency was performed for each analyte. Finally, the effect of the type of the employed calibration technique on the total variation of the method was examined. Calibration using the standard addition technique was proved more accurate for the determination in terms of analyte recovery. The sensitivity and recovery (Cd 99%, Co 101%, Cr 100%, Cu 99%, Mn 101%, Ni 100% and Pb 97%) of the developed method are presented for all examined blood fractions. Correspondence: George A. Zachariadis, Laboratory of Analytical Chemistry, Department of Chemistry, Aristotle University, Thessaloniki 54124, Greece     

气相色谱-场电离飞行时间质谱测定中间馏分油中链烷烃的形态分布

利用气相色谱-场电离飞行时间质谱技术( GC-FI TOF MS)建立了中间馏分油中不同异构程度链烷烃的碳数分布表征方法。首先利用GC-FI TOF MS技术对不同异构程度的链烷烃进行分离、鉴别,然后建立了中间馏分油沸点范围内正构烷烃、异构烷烃在GC-FI TOF MS测定时的相对响应因子算法及不同异构程度链烷烃的碳数分布算法,最后考察此定量方法的精密度和准确度。结果表明,GC-FI TOF MS可以将同碳数链烷烃区分为异构程度不同的3部分：多取代基异构烷烃、单取代基异构烷烃和正构烷烃;可利用GC-FI TOF MS对异构程度不同的链烷烃进行定量分析,精密度较好,相对偏差小于15%。此方法无需进行样品预分离,可直接进样分析,缩短了分析时间,且首次提供了不同异构程度异构烷烃的碳数分布信息。     

Continued fractions with multiple limits

For integers m2, we study divergent continued fractions whose numerators and denominators in each of the m arithmetic progressions modulo m converge. Special cases give, among other things, an infinite sequence of divergence theorems, the first of which is the classical Stern–Stolz theorem.We give a theorem on a class of Poincaré-type recurrences which shows that they tend to limits when the limits are taken in residue classes and the roots of their characteristic polynomials are distinct roots of unity.We also generalize a curious q-continued fraction of Ramanujan's with three limits to a continued fraction with k distinct limit points, k2. The k limits are evaluated in terms of ratios of certain q-series.Finally, we show how to use Daniel Bernoulli's continued fraction in an elementary way to create analytic continued fractions with m limit points, for any positive integer m2.