13 research outputs found

    The q-Exponential Operator and Generalized Rogers-Szego Polynomials

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    This paper is mainly concerned with using q-exponential operator T(bDq) in proving the identities that involve the generalized Rogers-Szego polynomials  rn(x,b) . We introduce some new roles of the q -exponential operator and prove that the generalized Rogers-Szego polynomials can be represented by theq -exponential operator, so we use this operator and it's roles in proving the basic identities rn(x,b)of  given in [7, 8] which are: generating function, Mehler's formula and Rogers formula. Then we introduce several extensions  of rn(x,b)  identities such that: the extended generating function, extended Mehler's formula, extended Rogers formula and another extended identities. These extended identities of the generalized Rogers-Szego polynomials can be considered a general form of the corresponding identities for the classical Rogers-Szego polynomials hn(x|q)  when b=1

    The generalized q-operator rɸs and its applications in q-identities

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    Based on the basic hypergeometric series rφs, we construct a new generalized q-operator rɸs (a1, . . . , ar b1, . . . , bs ; q, −cθ) and obtain some of its identities. Using these identities, we generalize several well-known q-identities, such as the q-Gauss sum, the q-Chu-Vandermonde sum, and the q-Pffaf-Saalschütz sum.Publisher's Versio

    On the Remes Algorithm for Rational Approximations

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    This paper is concerned with the minimax approximation of a discrete data set by rational functions. The second algorithm of Remes is applied. A crucial stage of this algorithm is solving the nonlinear system of leveling equations. In this paper, we will give a new approach for this purpose. In this approach, no initial guesses are required. Illustrative numerical example is presented

    Converging to Gosper's Algorithm

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    Given two polynomials, we find a convergence property of the GCD of the rising factorial and the falling factorial. Based on this property, we present a unified approach to computing the universal denominators as given by Gosper's algorithm and Abramov's algorithm for finding rational solutions to linear difference equations with polynomial coefficients.Comment: 13 page

    The Bivariate Rogers-Szeg\"{o} Polynomials

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    We present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers-Szeg\"{o} polynomials hn(x,yq)h_n(x,y|q). The proof of Mehler's formula can be considered as a new approach to the nonsymmetric Poisson kernel formula for the continuous big qq-Hermite polynomials Hn(x;aq)H_n(x;a|q) due to Askey, Rahman and Suslov. Mehler's formula for hn(x,yq)h_n(x,y|q) involves a 3ϕ2{}_3\phi_2 sum and the Rogers formula involves a 2ϕ1{}_2\phi_1 sum. The proofs of these results are based on parameter augmentation with respect to the qq-exponential operator and the homogeneous qq-shift operator in two variables. By extending recent results on the Rogers-Szeg\"{o} polynomials hn(xq)h_n(x|q) due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for hn(x,yq)h_n(x,y|q). Finally, we give a change of base formula for Hn(x;aq)H_n(x;a|q) which can be used to evaluate some integrals by using the Askey-Wilson integral.Comment: 16 pages, revised version, to appear in J. Phys. A: Math. Theo

    Impact of opioid-free analgesia on pain severity and patient satisfaction after discharge from surgery: multispecialty, prospective cohort study in 25 countries

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    Background: Balancing opioid stewardship and the need for adequate analgesia following discharge after surgery is challenging. This study aimed to compare the outcomes for patients discharged with opioid versus opioid-free analgesia after common surgical procedures.Methods: This international, multicentre, prospective cohort study collected data from patients undergoing common acute and elective general surgical, urological, gynaecological, and orthopaedic procedures. The primary outcomes were patient-reported time in severe pain measured on a numerical analogue scale from 0 to 100% and patient-reported satisfaction with pain relief during the first week following discharge. Data were collected by in-hospital chart review and patient telephone interview 1 week after discharge.Results: The study recruited 4273 patients from 144 centres in 25 countries; 1311 patients (30.7%) were prescribed opioid analgesia at discharge. Patients reported being in severe pain for 10 (i.q.r. 1-30)% of the first week after discharge and rated satisfaction with analgesia as 90 (i.q.r. 80-100) of 100. After adjustment for confounders, opioid analgesia on discharge was independently associated with increased pain severity (risk ratio 1.52, 95% c.i. 1.31 to 1.76; P < 0.001) and re-presentation to healthcare providers owing to side-effects of medication (OR 2.38, 95% c.i. 1.36 to 4.17; P = 0.004), but not with satisfaction with analgesia (beta coefficient 0.92, 95% c.i. -1.52 to 3.36; P = 0.468) compared with opioid-free analgesia. Although opioid prescribing varied greatly between high-income and low- and middle-income countries, patient-reported outcomes did not.Conclusion: Opioid analgesia prescription on surgical discharge is associated with a higher risk of re-presentation owing to side-effects of medication and increased patient-reported pain, but not with changes in patient-reported satisfaction. Opioid-free discharge analgesia should be adopted routinely

    Mortality from gastrointestinal congenital anomalies at 264 hospitals in 74 low-income, middle-income, and high-income countries: a multicentre, international, prospective cohort study

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    Summary Background Congenital anomalies are the fifth leading cause of mortality in children younger than 5 years globally. Many gastrointestinal congenital anomalies are fatal without timely access to neonatal surgical care, but few studies have been done on these conditions in low-income and middle-income countries (LMICs). We compared outcomes of the seven most common gastrointestinal congenital anomalies in low-income, middle-income, and high-income countries globally, and identified factors associated with mortality. Methods We did a multicentre, international prospective cohort study of patients younger than 16 years, presenting to hospital for the first time with oesophageal atresia, congenital diaphragmatic hernia, intestinal atresia, gastroschisis, exomphalos, anorectal malformation, and Hirschsprung’s disease. Recruitment was of consecutive patients for a minimum of 1 month between October, 2018, and April, 2019. We collected data on patient demographics, clinical status, interventions, and outcomes using the REDCap platform. Patients were followed up for 30 days after primary intervention, or 30 days after admission if they did not receive an intervention. The primary outcome was all-cause, in-hospital mortality for all conditions combined and each condition individually, stratified by country income status. We did a complete case analysis. Findings We included 3849 patients with 3975 study conditions (560 with oesophageal atresia, 448 with congenital diaphragmatic hernia, 681 with intestinal atresia, 453 with gastroschisis, 325 with exomphalos, 991 with anorectal malformation, and 517 with Hirschsprung’s disease) from 264 hospitals (89 in high-income countries, 166 in middleincome countries, and nine in low-income countries) in 74 countries. Of the 3849 patients, 2231 (58·0%) were male. Median gestational age at birth was 38 weeks (IQR 36–39) and median bodyweight at presentation was 2·8 kg (2·3–3·3). Mortality among all patients was 37 (39·8%) of 93 in low-income countries, 583 (20·4%) of 2860 in middle-income countries, and 50 (5·6%) of 896 in high-income countries (p<0·0001 between all country income groups). Gastroschisis had the greatest difference in mortality between country income strata (nine [90·0%] of ten in lowincome countries, 97 [31·9%] of 304 in middle-income countries, and two [1·4%] of 139 in high-income countries; p≤0·0001 between all country income groups). Factors significantly associated with higher mortality for all patients combined included country income status (low-income vs high-income countries, risk ratio 2·78 [95% CI 1·88–4·11], p<0·0001; middle-income vs high-income countries, 2·11 [1·59–2·79], p<0·0001), sepsis at presentation (1·20 [1·04–1·40], p=0·016), higher American Society of Anesthesiologists (ASA) score at primary intervention (ASA 4–5 vs ASA 1–2, 1·82 [1·40–2·35], p<0·0001; ASA 3 vs ASA 1–2, 1·58, [1·30–1·92], p<0·0001]), surgical safety checklist not used (1·39 [1·02–1·90], p=0·035), and ventilation or parenteral nutrition unavailable when needed (ventilation 1·96, [1·41–2·71], p=0·0001; parenteral nutrition 1·35, [1·05–1·74], p=0·018). Administration of parenteral nutrition (0·61, [0·47–0·79], p=0·0002) and use of a peripherally inserted central catheter (0·65 [0·50–0·86], p=0·0024) or percutaneous central line (0·69 [0·48–1·00], p=0·049) were associated with lower mortality. Interpretation Unacceptable differences in mortality exist for gastrointestinal congenital anomalies between lowincome, middle-income, and high-income countries. Improving access to quality neonatal surgical care in LMICs will be vital to achieve Sustainable Development Goal 3.2 of ending preventable deaths in neonates and children younger than 5 years by 2030

    The q-Exponential Operator

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    Abstract We define a q-exponential operator R(bD q ) which turn out to be suitable for dealing with the Cauchy polynomials P n (x, y) and the homogeneous Rogers-Szegö polynomials h n (x, y|q). By using this operator, we derive Mehler&apos;s formula and Rogers formula for the polynomials P n (x, y) and h n (x, y|q). Mathematics Subject Classification: 05A30, 33D4

    Generalized q-difference equation of the generalized q-operator rΦs(θ) and its application in q-integrals

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    In this paper, we employ the q-difference equation technique to generalize some well-known q-integrals such as the extension of Askey–Roy q-integral, Andrews–Askey q-integral, and Askey–Wilson q-integral
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