We study extension of p-trigonometric functions sinp and cosp to complex domain. For p = 4, 6, 8, . . ., the function sinp satisfies the initial value problem which is equivalent to (∗) −(u ′ ) p−2 u ′′ − u p−1 = 0, u(0) = 0, u ′ (0) = 1 in R. In our recent paper, Girg, Kotrla (2014), we showed that sinp(x) is a real analytic function for p = 4, 6, 8, . . . on (−πp/2, πp/2), where πp/2 = R 1 0 (1 − s p ) −1/p. This allows us to extend sinp to complex domain by its Maclaurin series convergent on the disc {z ∈ C: |z| < πp/2}. The question is whether this extensions sinp(z) satisfies (∗) in the sense of differential equations in complex domain. This interesting question was posed by Došlý and we show that the answer is affirmative. We also discuss the difficulties concerning the extension of sinp to complex domain for p = 3, 5, 7, . . . Moreover, we show that the structure of the complex valued initial value problem (∗) does not allow entire solutions for any p ∈ ℕ, p > 2. Finally, we provide some graphs of real and imaginary parts of sinp(z) and suggest some new conjectures.
Two new, one known and three unidentified species of the nematode family Camallanidae are reported and described from the intestines of marine perciform fishes off the southwestern coast of New Caledonia, South Pacific: Camallanus carangis Olsen, 1952 from the forked-tailed threadfin bream Nemipterus furcosus (Nemipteridae), the yellowstriped goatfish Upeneus vittatus and the whitesaddle goatfish Parupeneus ciliatus (both Mullidae) (new host records); Procamallanus (Spirocamallanus) variolae sp. n. from the white-edged lyretail Variola albimarginata (type host) and the blacktip grouper Epinephelus fasciatus (both Serranidae); Procamallanus (Spirocamallanus) longus sp. n. from the twotone tang Zebrasoma scopas (Acanthuridae); Procamallanus (Spirocamallanus) sp. 1 (female tail with 2 terminal spikes on a digit-like projection) from the speckled sandperch Parapercis hexophtalma (Pinguipedidae); Procamallanus (Spirocamallanus) sp. 2 (female tail with 1 spike on a digit-like projection) from the drab emperor Lethrinus ravus (Lethrinidae) and Procamallanus (Spirocamallanus) sp. 3 (female tail with a smooth digit-like protrusion) from the two-lined monocle bream Scolopsis bilineata (Nemipteridae). Camallanus paracarangis Velasquez, 1980 is synonymized with C. carangis. Several additional species of Camallanus from marine fish of the Indo-Pacific region may be synonymous with C. carangis as it has a poorly sclerotized left spicule and 3 small caudal projections on the tail of young (i.e., non-gravid) females. The fourth-stage larva of C. carangis is described for the first time. P. (S.) variolae differs from most similar species of this region mainly in the position (i.e., at level or posterior to the nerve ring) and shape of deirids. P. (S.) longus differs from the similar P. (S.) chaimha mainly in a different arrangement of postanal papillae, shape of the female tail, much longer right spicule (429 µm) and longer body of gravid females (38-55 mm). All Camallanus and Procamallanus (Spirocamallanus) spp. reported here represent the first records of camallanids from marine fishes in New Caledonian waters.