In [17], Changhee-Genocchi polynomials are defined by the generating function to be begin{aligned} frac{2 log(1+t)}{2+t} (1+t)^{x} = sum_{n=0}^{infty}mathit{CG}_{n}(x) frac{t^{n}}{n!}.
end{aligned} Here, (E_{n}(x)) are ordinary Euler polynomials which are given by the generating function to be begin{aligned} frac{2}{e^{t}+1}e^{xt} = sum _{n=0}^{infty}E_{n}(x) frac{t^{n}}{n!}.
The modified degenerate Changhee-Genocchi polynomials are considered by the generating function to be begin{aligned} frac{2tlambda}{2lambda+log(1+lambda t)} bigl(1+ lambda^{-1}log (1+lambda t) bigr)^{x} = sum _{n=0}^{infty}mathit{CG}_{n,lambda}^(x) frac{t^{n}}{n!}.
Recently, the Changhee polynomials have been defined by the generating function to be begin{aligned} frac{2}{t+2}(1+t)^{x} = sum_{n=0}^{infty}mathit{Ch}_{n}(x) frac{t^{n}}{n!}quad bigl text{see [9, 11--17]}bigr).
First, we consider the Changhee-Genocchi polynomials of the second kind which are given by the generating function to be begin{aligned} frac{2log(1+t)}{ (1+lambdalog(1+t) )^{frac{1}{lambda }+11} bigl(1+lambda log(1+t) bigr)^{frac{x}{lambda}}= sum_{n=0}^{infty}J_{n,lambda}(x) frac{t^{n}}{n!}.
As is known, the Genocchi polynomials are defined by the generating function to be begin{aligned} frac{2t}{e^{t}+1}e^{xt} = sum _{n=0}^{infty}G_{n}(x) frac{t^{n}}{n!}quad bigl text{see [1--25]}bigr).
In [17], the degenerate Changhee-Genocchi polynomials are defined by the generating function to be begin{aligned} frac{2lambdalog(1+ frac{1}{lambda}log (1+lambda t) ) }{2lambda+log(1+lambda t)} bigl( 1+ lambda^{-1}log(1+lambda t) bigr)^{x} = sum _{n=0}^{infty}mathit{CG}_{n,lambda}(x) frac{t^{n}}{n!}.
The degenerate Changhee polynomials of the second kind are also defined by the generating function to be begin{aligned} frac{2}{ (1+lambdalog(1+t) )^{frac{1}{lambigl}+1+lambda(1+logbda log(1+t) bigr)^{frac{x}{lambda}} = sum_{n=0}^{infty}C_{n,lambda}(x) frac{t^{n}}{n!}quadbigl text{see [8]}bigr).
As is well known, the Bernoulli numbers of the second kind are defined by the generating function to be begin{aligned} frac{t}{log(1+t)} = sum _{n=0}^{infty}b_{n} frac{t^{n}}{n!}quad bigl text{see} text{[24]}bigr).
Using this idea, they obtain the normalized function to be begin{aligned} M alpha )=frac{2}{2-alpha }, quad 0le alpha le 1. end{aligned}Therefore, the Caputo Fabrizio fractional derivative of order (0<alpha <1) was reformulated by Losada and Nieto as begin{aligned} ^{rm CF}D^{alpha }u(x)=frac{1}{1-alpha }int _0^x exp left( frac{-alpha (x-t)}{1-alpha }right) u^(t) {rm d}t.
Condition for a function to be concave, begin{array}{*{20}l} Delta_{1}leq0 Dend{arrayeq0 end{array}.
