The Reversed q-Exponential Functional Relation

After obtaining some useful identities, we prove an additional functional relation for $q$ exponentials with reversed order of multiplication, as well as the well known direct one in a completely rigorous manner.Comment: 6 pages, LaTeX, no figure

On Fourier integral transforms for qq-Fibonacci and qq-Lucas polynomials

We study in detail two families of $q$-Fibonacci polynomials and $q$-Lucas polynomials, which are defined by non-conventional three-term recurrences. They were recently introduced by Cigler and have been then employed by Cigler and Zeng to construct novel $q$-extensions of classical Hermite polynomials. We show that both of these $q$-polynomial families exhibit simple transformation properties with respect to the classical Fourier integral transform

h analogue of Newton's binomial formula

In this letter, the $h$--analogue of Newton's binomial formula is obtained in the $h$--deformed quantum plane which does not have any $q$--analogue. For $h=0$, this is just the usual one as it should be. Furthermore, the binomial coefficients reduce to $\frac{n!}{(n-k)!}$ for $h=1$. \\ Some properties of the $h$--binomial coefficients are also given. \\ Finally, I hope that such results will contribute to an introduction of the $h$--analogue of the well--known functions, $h$--special functions and $h$--deformed analysis.Comment: 6 pages, latex Jounal-ref: J. Phys. A: Math. Gen. 31 (1998) L75

Exactly solvable D_N-type quantum spin models with long-range interaction

We derive the spectra of the D_N-type Calogero (rational) su(m) spin model, including the degeneracy factors of all energy levels. By taking the strong coupling limit of this model, in which its spin and dynamical degrees of freedom decouple, we compute the exact partition function of the su(m) Polychronakos-Frahm spin chain of D_N type. With the help of this partition function we study several statistical properties of the chain's spectrum, such as the density of energy levels and the distribution of spacings between consecutive levels.Comment: 12 pages, 2 figure

Central factorials under the Kontorovich-Lebedev transform of polynomials

We show that slight modifications of the Kontorovich-Lebedev transform lead to an automorphism of the vector space of polynomials. This circumstance along with the Mellin transformation property of the modified Bessel functions perform the passage of monomials to central factorial polynomials. A special attention is driven to the polynomial sequences whose KL-transform is the canonical sequence, which will be fully characterized. Finally, new identities between the central factorials and the Euler polynomials are found.Comment: also available at http://cmup.fc.up.pt/cmup/ since the 2nd August 201

Relationship between sensor-detected Hypoglycemia and patient-reported hypoglycemia in people with type 1 and insulin-treated type 2 diabetes: The Hypo-METRICS study

OBJECTIVE Use of continuous glucose monitoring (CGM) has led to greater detection of hypoglycemia; the clinical significance of this is not fully understood. The Hypoglycaemia–Measurement, Thresholds and Impacts (Hypo-METRICS) study was designed to investigate the rates and duration of sensor-detected hypoglycemia (SDH) and their relationship with person-reported hypoglycemia (PRH) in people living with type 1 diabetes (T1D) and insulin-treated type 2 diabetes (T2D) with prior experience of hypoglycemia. RESEARCH DESIGN AND METHODS We recruited 276 participants with T1D and 321 with T2D who wore a blinded CGM and recorded PRH in the Hypo-METRICS app over 10 weeks. Rates of SDH <70 mg/dL, SDH <54 mg/dL, and PRH were expressed as median episodes per week. Episodes of SDH were matched to episodes of PRH that occurred within 1 h. RESULTS Median [interquartile range] rates of hypoglycemia were significantly higher in T1D versus T2D; for SDH <70 mg/dL (6.5 [3.8–10.4] vs. 2.1 [0.8–4.0]), SDH <54 mg/dL (1.2 [0.4–2.5] vs. 0.2 [0.0–0.5]), and PRH (3.9 [2.4–5.9] vs. 1.1 [0.5–2.0]). Overall, 65% of SDH <70 mg/dL was not associated with PRH, and 43% of PRH had no associated SDH. The median proportion of SDH associated with PRH in T1D was higher for SDH <70 mg/dL (40% vs. 22%) and SDH <54 mg/dL (47% vs. 25%) than in T2D. CONCLUSIONS The novel findings are that at least half of CGM hypoglycemia is asymptomatic, even below 54 mg/dL, and many reported symptomatic hypoglycemia episodes happen above 70 mg/dL. In the clinical and research setting, these episodes cannot be used interchangeably, and both need to be recorded and addressed