Differential Equation of Appell Polynomials Via the Factorization Method

Let {Pn(x)}∞n=0 be a sequence of polynomials of degree n. We define two sequences of differential operators Φn and Ψn satisfying the following properties: By constructing these two operators for Appell polynomials, we determine their differential equations via the factorization method introduced by Infeld and Hull (Rev. Mod. Phys. 23 (1951) 21). The differential equations for both Bernoulli and Euler polynomials are given as special cases of the Appell polynomials

Monomiality Principle and Eigenfunctions of Differential Operators

We apply the so-calledmonomiality principlein order to construct eigenfunctions for a wide set of ordinary differential operators, relevant to special functions and polynomials, including Bessel functions and generalized Gould-Hopper polynomials

Adjoint Appell-Euler and First Kind Appell-Bernoulli Polynomials

The adjunction property, recently introduced for Sheffer polynomial sets, is considered in the case of Appell polynomials. The particular case of adjoint Appell-Euler and Appell-Bernoulli polynomials of the first kind is analyzed

Asymptotics of Zeros of Relativistic Hermite Polynomials

The relativistic Hermite polynomial (RHP) is a class of orthogonal polynomials associated with varying weights. We study the asymptotics of the zeros of the RHP when both degree $n$ of polynomials and relativistic parameter $N$ approach infinity

On Quadrature Rules Associated with Appell Polynomials

A quadrature rule using Appell polynomials and generalizing both the Euler-MacLaurin quadrature formula and a similar quadrature rule, obtained in Bretti et al [15], which makes use of Euler (instead of Bernoulli) numbers and even (instead of odd) derivatives of the given function at the extrema of the considered interval, is derived. An expression of the remainder term and a numerical example are also enclosed

IN MEMORY OF GAETANO FICHERA

Sono raccolte in questo Volume alcune delle conferenze svolte durante il 3^o Simposio Internazionale Problemi Attuali dell’Analisi e della Fisica Matematica, dedicato alla memoria di Gaetano Fichera, che si è svolto a Taormina dal 29 Giugno al 1 Luglio, 2006. Hanno partecipato al Simposio più di cinquanta studiosi italiani e stranieri (cfr. Elenco dei Partecipanti). Sono state svolte trentasette relazioni.Sono raccolte in questo Volume alcune delle conferenze svolte durante il3o Simposio Internazionale Problemi Attuali dell’Analisi e della Fisica Matematica, dedicato alla memoria di Gaetano Fichera, che si è svolto a Taormina dal 29 Giugno al 1 Luglio, 2006. Hanno partecipato al Simposio più di cinquanta studiosi italiani e stranieri (cfr. Elenco dei Partecipanti). Sono state svolte trentasette relazioni

MULTI-VARIABLE GOULD-HOPPER AND LAGUERRE POLYNOMIALS

The monomiality principle was introduced by G. Dattoli, in order to derive the properties of special or generalized polynomials starting from the corresponding ones of monomials. In this article we show a general technique to extend themonomiality approach tomulti-index polynomials in several variables. Application to the case of Hermite, Laguerre-type and mixed-type (i.e. between Laguerre and Hermite) are derived.The monomiality principle was introduced by G. Dattoli, in order toderive the properties of special or generalized polynomials starting fromthe corresponding ones of monomials. In this article we show a generaltechnique to extend themonomiality approach tomulti-index polynomials in several variables. Application to the case of Hermite, Laguerre-type and mixed-type (i.e. between Laguerre and Hermite) are derived

Is contrast-enhanced US alternative to spiral CT in the assessment of treatment outcome of radiofrequency ablation in hepatocellular carcinoma?

Purpose: The present study was conducted to assess the efficacy of contrast-enhanced ultrasound with low mechanical index in evaluating the response of percutaneous radiofrequency ablation treatment of hepatocellular carcinoma by comparing it with 4-row spiral computed tomography. Materials and Methods: 100 consecutive patients (65 men and 35 women; age range: 62 – 76 years) with solitary hepatocellular carcinomas (mean lesion diameter: 3.7cm± 1.1cm SD) underwent internally cooled radiofrequency ablation. Therapeutic response was evaluated at one month after the treatment with triple-phasic contrast-enhanced spiral CT and low-mechanical index contrast-enhanced ultrasound following bolus injection of 2.4 ml of Sonovue (Bracco, Milan). 60 out of 100 patients were followed up for another 3 months. Contrast-enhanced sonographic studies were reviewed by two blinded radiologists in consensus. Sensitivity, specificity, NPV and PPV of contrast-enhanced ultrasound examination were determined. Results: After treatment, contrast-enhanced ultrasound identified persistent signal enhancement in 24 patients (24%), whereas no intratumoral enhancement was detected in the remaining 76 patients (76%). Using CT imaging as gold standard, the sensitivity, specificity, NPV, and PPV of contrast enhanced ultrasound were 92.3% (95% CI = 75.9 – 97.9%), 100% (95% CI = 95.2 – 100%), 97.4% (95% CI = 91.1 – 99.3%), and 100% (95% CI = 86.2 – 100%). Conclusion: Contrast-enhanced ultrasound with low mechanical index using Sonovue is a feasible tool in evaluating the response of hepatocellular carcinoma to radiofrequency ablation. Accuracy is comparable to 4-row spiral CT

Cortical responses to salient nociceptive and not nociceptive stimuli in vegetative and minimal conscious state

Aims: Questions regarding perception of pain in non-communicating patients and the management of pain continue to raise controversy both at a clinical and ethical level. The aim of this study was to examine the cortical response to salient visual, acoustic, somatosensory electric non-nociceptive and nociceptive laser stimuli and their correlation with the clinical evaluation. Methods: Five Vegetative State (VS), 4 Minimally Conscious State (MCS) patients and 11 age- and sex-matched controls were examined. Evoked responses were obtained by 64 scalp electrodes, while delivering auditory, visual, non-noxious electrical and noxious laser stimulation, which were randomly presented every 10 s. Laser, somatosensory, auditory and visual evoked responses were identified as a negative-positive (N2-P2) vertex complex in the 500 ms post-stimulus time. We used Nociception Coma Scale-Revised (NCS-R) and Coma Recovery Scale (CRS-R) for clinical evaluation of pain perception and consciousness impairment. Results: The laser evoked potentials (LEPs) were recognizable in all cases. Only one MCS patient showed a reliable cortical response to all the employed stimulus modalities. One VS patient did not present cortical responses to any other stimulus modality. In the remaining participants, auditory, visual and electrical related potentials were inconstantly present. Significant N2 and P2 latency prolongation occurred in both VS and MCS patients. The presence of a reliable cortical response to auditory, visual and electric stimuli was able to correctly classify VS and MCS patients with 90% accuracy. Laser P2 and N2 amplitudes were not correlated with the CRS-R and NCS-R scores, while auditory and electric related potentials amplitude were associated with the motor response to pain and consciousness recovery. Discussion: pain arousal may be a primary function also in vegetative state patients while the relevance of other stimulus modalities may indicate the degree of cognitive and motor behavior recovery. This underlines the importance of considering the potential experience of pain also in patient

Effect of thyroglobulin autoantibodies on the metabolic clearance of serum thyroglobulin

Background: In order to establish whether thyroglobulin autoantibodies (TgAb) influence the metabolic clearance of thyroglobulin (Tg) in humans, serum Tg and TgAb were correlated shortly after radioiodine (131I) treatment. Methods: Samples were collected from 30 consecutive patients undergoing 131I activity for Graves' hyperthyroidism at the time of treatment and every 15 days thereafter, up to 90 days. Tg and TgAb were measured by immunometric assays (functional sensitivities: 0.1 ng/mL and 8 IU/mL). Results: Tg was detectable in all patients at day 0. Tg concentrations rose from a mean of 33.2 ng/mL [confidence interval (CI) 17.8–61.0 ng/mL] at day 0 to a mean of 214.6 ng/mL [CI 116.9–393.4 ng/mL] at day 30 and then steadily decreased, reaching the lowest concentration at day 90 (M = 10.9 ng/mL [CI 5.5–20.9 ng/mL]). Compared to their levels at day 0 (M = 23.6 IU/mL [CI 10.5–52.9 IU/mL]), TgAb remained stable through day 15 and then gradually increased up to a mean of 116.6 IU/mL [CI 51.9–262.2 IU/mL] at day 90. Patients were then split into two groups according to their TgAb status at day 0: undetectable (<8 IU/mL; 9 patients) or detectable (≥8 IU/mL; 21 patients) TgAb. Compared to the other cohort, patients with detectable TgAb showed significantly lower Tg concentrations at day 0 (M = 20.3 ng/mL [CI 10.1–40.2 ng/mL] vs. M = 101.8 ng/mL [CI 36.6–279.8 ng/mL]), similar at day 15, lower levels at day 30 (M = 146.5 ng/mL [CI 74.3–287.8 ng/mL] vs. M = 514.8 ng/mL [CI 187.8–1407.9 ng/mL]), at day 45 (M = 87.5 ng/mL [CI 43.1–176.6 ng/mL] vs. M = 337.9 ng/mL [CI 120.1–947.0 ng/mL]), at day 60 (M = 61.6 ng/mL [CI 31.0–121.4 ng/mL] vs. M = 255.8 ng/mL [CI 79.0–823.8 ng/mL]), and at day 75 (M = 24.5 ng/mL [CI 11.9–49.2 ng/mL] vs. M = 249.5 ng/mL [CI 63.5–971.1 ng/mL]), and similar levels at day 90. Patients with detectable TgAb showed a lower (M = 182.5 ng/mL [CI 92.0–361.0 ng/mL] vs. M = 514.8 ng/mL [CI 187.8–1407.9 ng/mL]) and an earlier (day 15 vs. day 30) peak of Tg. The mean Tg concentration was lower in patients with detectable TgAb than in those with undetectable TgAb (area under the curve: 17,340 ± 16,481 ng/mL vs. 36,883 ± 44,625 ng/mL; p = 0.02). Conclusions: TgAb influence the changes in Tg concentrations observed immediately after 131I treatment, inducing lower levels and an earlier peak of Tg. These observations indicate that TgAb significantly influence the metabolic clearance of Tg, supporting the concept that their interference in the measurement of Tg is mainly due to an in vivo effect