Abstract
This paper studies an extended Bessel function of the form
B-a(b,p,c)(x) := Sigma(infinity)(k=0)(-c)(k)/k!Gamma(ak + p + b+1/2) (x/2)(2k+p).
Representation formulations for B-a(b,p,c) are derived in terms of the parameters a, b, and p. An important consequence is the derivation of an (a + 1)-order differential equation satisfied by the function B-a(b,p,c) . Interesting functional inequalities are established, particularly for the case a = 2, and c = +/-alpha(2).
Monotonicity properties of B-a(b,p,c) are also studied for non-positive c. Log-concavity and log-convexity properties in terms of the parameters d and p are respectively investigated for the closely related function
B-a(b,p,c)d(x) := Sigma(infinity)(k=0)(-c/4)(k)Gamma(p + b+1/2)/Gamma( k + 1)Gamma(ak + p + b+1/2) (d)(k)/k!x(k),
which leads to direct and reverse Turan-type inequalities.