Baran ρυ(x) function

Title: The ϱυ Function: A New Special Function from the Lame Curve Area IntegralDescription:We introduce and study the function ϱυ(x) defined by the integral of (1 minus t to the power x) to the power (1/x), integrated from 0 to 1, for x greater than 0. Its closed form is the square of the Gamma function evaluated at (1 plus 1/x), divided by the Gamma function evaluated at (1 plus 2/x).Geometrically, ϱυ(x) measures the area of the first-quadrant region enclosed by the Lame curve t^x plus s^x equals 1. As x increases from 0 to infinity, this region continuously interpolates between a triangle (x=1, area=1/2), a quarter circle (x=2, area=pi/4), and the unit square (x→∞, area=1).Five complete series expansions are derived: a binomial series valid for all x greater than 0; a Taylor expansion around x=1 with the closed coefficient c2=(1−ζ(2))/2; a logarithmic asymptotic series in terms of Riemann zeta values ζ(2), ζ(3), ζ(4), ...; an exact formula at reciprocal integers ϱυ(1/n) = 1/C(2n,n), connecting the function to the central binomial coefficients; and a Stirling-type estimate as x approaches 0.Additional results include: the derivative formula in terms of the digamma function; the product identity ϱυ(2) times ϱυ(1/2) equals pi/24; the inverse function and its Puiseux expansion near y=1; and the connection to the Baran Area Function A_p(x) via the identity ϱυ(p) = A_p(1).This function does not appear to have been previously named or systematically studied in the literature, despite its natural connection to the geometry of Lame curves and the Beta function.Keywords: special function, Lame curve, superellipse, area integral, central binomial coefficients, Riemann zeta function, Beta function, Gamma function, Puiseux series, Baran area functionMSC: 33E99, 26A48, 26B15 Abdullah Baran, van Çatak