A Note on the Summation of the Incomplete Gamma Function

We examine the improved infinite sum of the incomplete gamma function for large values of the parameters involved. We also evaluate the infinite sum and equivalent Hurwitz-Lerch zeta function at special values and produce a table of results for easy reading. Almost all Hurwitz-Lerch zeta functions have an asymmetrical zero distribution.

Energy analysis of Duffing oscillators with quadratic damping: exact solutions

Oscillators with a Duffing-type restoring force and quadratic damping are dealt with in this paper. Four characteristic cases of this restoring force are analysed: hardening, softening, bistable and a pure cubic one. Their energy-displacement relationships are considered, and the corresponding closed-form exact solutions are obtained in terms of the incomplete Gamma function, which represent new results. Such results provide insight into damped dynamics of the class of system, including finding the phase trajectories as well as the comparison between these cases from the viewpoint of the energy loss per cycle.

Poisson like matrix operator and its application in p-summable space

The incomplete gamma function Γ(a, u) is defined by Γ ( a , u ) = ∫ u ∞ t a − 1 e − t d t , $$\Gamma(a,u)=\int\limits_{u}^{\infty}t^{a-1}\textrm{e}^{-t}\textrm{d} t,$$ where u > 0. Using the incomplete gamma function, we define a new Poisson like regular matrix P ( μ ) = ( p n k μ ) $\mathfrak{P}(\mu)=(p^{\mu}_{nk})$ given by p n k μ = n ! Γ ( n + 1 , μ ) e − μ μ k k ! ( 0 ≤ k ≤ n ) , 0 ( k > n ) , $$p^{\mu}_{nk}= \begin{cases} \dfrac{n!}{\Gamma(n+1,\mu)}\dfrac{\textrm{e}^{-\mu}\mu^k}{k!} \quad &(0\leq k\leq n), \\[1ex] 0\quad & (k>n), \end{cases}$$ where μ > 0 is fixed. We introduce the sequence space ℓ p ( P ( μ ) ) $\ell_p(\mathfrak{P}(\mu))$ for 1 ≤ p ≤ ∞ and some topological properties, inclusion relations and generalized duals of the newly defined space are discussed. Also we characterize certain matrix classes and compact operators related to the space ℓ p ( P ( μ ) ) $\ell_p(\mathfrak{P}(\mu))$ . We obtain Gurarii’s modulus of convexity and investigate some geometric properties of the new space. Finally, spectrum of the operator P ( μ ) $\mathfrak{P}(\mu)$ on sequence space c 0 has been investigated.

A HARMONIC MEAN INEQUALITY CONCERNING THE GENERALIZED EXPONENTIAL INTEGRAL FUNCTION

In this paper, we prove that for $s\in(0,\infty)$, the harmonic mean of $E_k(s)$ and $E_k(1/s)$ is always less than or equal to $\Gamma(1-k,1)$. Where $E_k(s)$ is the generalized exponential integral function, $\Gamma(u,s)$ is the upper incomplete gamma function and $k\in \mathbb{N}$.

Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function

We apply our simultaneous contour integral method to an infinite sum in Prudnikov et al. and use it to derive the infinite sum of the Incomplete gamma function in terms of the Hurwitz zeta function. We then evaluate this formula to derive new series in terms of special functions and fundamental constants. All the results in this work are new.

Lévy Processes Linked to the Lower-Incomplete Gamma Function

We start by defining a subordinator by means of the lower-incomplete gamma function. This can be considered as an approximation of the stable subordinator, easier to be handled in view of its finite activity. A tempered version is also considered in order to overcome the drawback of infinite moments. Then, we study Lévy processes that are time-changed by these subordinators with particular attention to the Brownian case. An approximation of the fractional derivative (as well as of the fractional power of operators) arises from the analysis of governing equations. Finally, we show that time-changing the fractional Brownian motion produces a model of anomalous diffusion, which exhibits a sub-diffusive behavior.

Lactation curve, milk production of Pelibuey ewes and preweaning growth rate of the lambs

Objective: To estimate the lactation curve and milk production of Pelibuey ewes andthe relationship with preweaning growth rate of the lambs.Design/methodology/approach: Forty five Pelibuey ewes were milked during 70days in Montecillo, México, in 2018, to estimate daily and total milk production. Thelactation curve was fitted with the incomplete gamma function. In addition, the effectsof type of birth and ewe weight at milking on milk production were analyzed, andcorrelations were calculated between ewe milk production and growth rate of thelambs, per week and for the entire lactation Results: A “typical” lactation curve was found, average ewe milk production for theentire lactation, weighted for the number of lambs suckling, was 131±8 L, with444±24 g d -1 . Ewe weight at milking had an effect (p<0.01) on milk production.Positive correlations were found (p<0.05) between ewe milk production andpreweaning growth rate of the lambs.Limitations on study/implications: There is a strong dependency of the lambs forthe milk production of the Pelibuey ewe, a factor of great relevance so that lambs cangain body weight and survive during lactation.Findings/conclusions: Pelibuey ewes produce less milk than dairy ewes. Therefore,lambs should be weaned at a maximum of 10 weeks of lactation.