On the zeros of the hypergeometric function

Bounded analytic functions in the unit disk and the behaviour of certain analytic continued fractions near singular line.

“…[3], Theorem): Now F(a, &, c; z) and F(a + l, δ, c + 1; z) do not have any common zeros in C* because of a(c -b)F(a + l, 6, c + 1; z) -acF(a, b, c; z) + (l -z)cF'(a, b, c;z) =0 and because of the hypergeometric differential equation.…”

“…p. 339, (89.14), and where F(a,b,c;z) denotes the principal branch of the hypergeometric function in C*. 5 F(a, b, c;z) According to [3] this also can be deduced from the hypergeometric differential equation. .…”

“…[3], Ί -^IOTI / p. 53, Lemma l, and observe K (t) = -~-= -there) that Next (47) implies N(f n ) = P(f n +i) f( > r a11 n · Using this and (53) with n + l we obtain from (52) and (53) l (54) P n -P n+2 = l -T (sign a n A n + sign ^n +1 ) sign Im f n+l (z) if (50) holds. Then it is a simple consequenee of the argument principle (cf.…”

“…and where F(a, b, c;z) is the hypergeometric function with real α, b, c. That in this case the conclusion of Wall's Theorem is true can be seen for instance from [3]. ___ F(a,b,c;z) ' * n 2 2 c + 2n ' 2n+1 2 2 c + 2n + l for n = 0, l, 2, .…”

Completely monotone sequences and universally prestarlike functions

“…One should remark that there are a number of theorems by various authors (for instance, Van Vleck [21], Hurwitz [8], Schafheitlin [17], Runckel [14] and Küstner [9]) dealing with the non-vanishing of hypergeometric functions in Λ. Our results above concern univalence in Λ, which, at least for the derivatives, imply non-vanishing statements as well, and the corresponding parameter sets {a, b, c} have large intersections.…”