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We give a systematic summary of the properties of the Gegenbauer functions Cαλ(x) and Dαλ(x) for general complex degree and order, with emphasis on the functions of the second kind, Dαλ(x), and on results useful in scattering theory. The results presented include Sommerfeld–Watson type expansion formulas and two reciprocal addition formulas for the functions of the second kind.

We propose that the relative cross sections for the reaction processes r\p,r\ 'p^^ rip,ri'p,K^h and K~p-^ r\n,r]n, induced by pseudoscalar mesons on a proton target, provide a sensitive test for the presence of a strange-antistrange (55) quark component in the nucleon's wave function.PACS numbers: l2.40.Aa, 14.2aDh In the naive quark model, the structure of the proton is |p)= \uud). The question of the quark content has been raised again by recent analyses of measurements' of the polarized structure function of the proton. Some authors^ argue that these data imply a significant admixture of strange-antistrange iss) pairs in the nucleon, even for small momentum transfers (large distances), while others^ dispute this interpretation. The 55 content in the nucleon can also be probed in low-energy pionnucleon scattering"* (via the so-called "cr term") or in nucleon-antinucleon (A^A^) annihilation processes. In the latter case, it has been suggested^ that the 55"component is revealed in terms of a breakdown of the OkuboZweig-Iizuka (OZI) rule at some level. This interpretation has been questioned.^ According to the OZI rule,^ for example, the ratio of cross sections G(A-^B-* (p +X)/G{A +5-(o-^-X), where A, B, and X do not contain strange quarks, is of order 1%. The dynamical origin of the OZI rule has been recently discussed by Lipkin.^ Tests of the type mentioned above involve an amplitude proportional to the square of the ss component of the nucleon wave function.In this Letter, we propose a first-order test of the strange-quark content of the nucleon. Specifically, we consider the pseudoscalar meson-baryon reactions K~p-^ rin.rj'n and r\p,ri'p-* rjp.rj'p.K'^A. We focus on the T] and 77' , since these mesons contain significant components of both strange (rjs =ss/^) and nonstrange lriud^(tiu-dd)/2] quark-antiquark (QQ) configurations. Because of this fortunate circumstance, interference effects between the 77^ and rfud components could be substantial in the reaction amplitudes. These amplitudes contain a term linear in the strength C^^ of the ss component of the proton, written schematically as I /?) == CI uud) + Css I uudss) -f(1)In contrast, the amplitude for Kp-^ np is quadratically dependent on Q^, since the pion has no ss component.An interference effect in r\,r] reactions arises because the relative phase of the r]ud and r\s components is opposite in the 77 and r\. Explicitly, we have the flavor wave functions ri=a7]ud-pr]s , ri'=Pr]ud-^ar]s , where we have ignored a possible gluonic component of the 77' . Our normalization is thus a^H-j3^=2. In terms of the conventional mixing angle 0, defined by (2) 9-11 77=cos^|8)-sin0| 1), (3) 77'=sin0|8) + cos^| l>, where \S}^(uu-\-dd-2ss)/^ and |l)==(ww+rfj -\-ss)/y/3 are SU(3) octet and singlet combinations, respectively, we have a = P = 1/2 COS0 -2sin6> V3 1/2 sin^-f 2cos^ V3Most recent decay analyses'^ suggest 0^ -20° which a^ 1.162,^=^^0.806. We now define the following amplitudes: A^<'r]udp\T\T]udP>. B=-{T],P\ T\T]SP} , C=^{r]udp\ T\T]SP) .(4)for (5) Some quark ...

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