By looking under the hood of Zeilberger’s algorithm, as simplified by Mohammed and Zeilberger, it is shown that all the classical hypergeometric closed-form evaluations can be discovered ab initio, as well as many “strange” ones of Gosper, Maier, and Gessel and Stanton. The accompanying Maple package FindHypergeometric explains the various miracles that account for the classical evaluations, an...

We compute the exact and limiting smallest eigenvalue distributions for two classes of β-Jacobi ensembles not covered by previous studies. In the general β case, these distributions are given by multivariate hypergeometric 2F1 2/β functions, whose behavior can be analyzed asymptotically for special values of β which include β ∈ 2N+ as well as for β = 1. Interest in these objects stems from thei...

We investigate the approximation of some hypergeometric functions of two variables, namely the Appell functions Fi, i = 1, . . . , 4, by multivariate Padé approximants. Section 1 reviews the results that exist for the projection of the Fi onto x = 0 or y = 0, namely, the Gauss function 2F1(a, b; c; z), since a great deal is known about Padé approximants for this hypergeometric series. Section 2...

Inhibitory lectin-like receptors expressed on the surface of hematopoietic cells are critically involved in regulation of their effector functions. Here we report that a novel mAb specific for mouse NK cells, 2F1, recognizes the mouse homolog of the mast cell function-associated antigen (MAFA), an inhibitory lectin-like transmembrane receptor expressed on rat mast cells. The 2F1 antigen (2F1-Ag...

the object of this paper is to establish certain generalized fractional integration and differentiation involving generalized mittag-leffler function defined by salim and faraj [25]. the considered generalized fractional calculus operators contain the appell's function $f_3$ [2, p.224] as kernel and are introduced by saigo and maeda [23]. the marichev-saigo-maeda fractional calculus operat...

We investigate the zeros of a family of hypergeometric polynomials Mn(x;β, c) = (β)n 2F1(−n,−x;β; 1 − 1c ), n ∈ N, known as Meixner polynomials, that are orthogonal on (0,∞) with respect to a discrete measure for β > 0 and 0 < c < 1. When β = −N , N ∈ N and c = p p−1 , the polynomials Kn(x; p,N) = (−N)n 2F1(−n,−x;−N ; 1 p ), n = 0, 1, . . . N , 0 < p < 1 are referred to as Krawtchouk polynomial...

Each family of Gauss hypergeometric functions fn = 2F1(a + ε1n, b + ε2n; c + ε3n; z), n ∈ Z , for fixed εj = 0,±1 (not all εj equal to zero) satisfies a second order linear difference equation of the form Anfn−1 + Bnfn + Cnfn+1 = 0. Because of symmetry relations and functional relations for the Gauss functions, many of the 26 cases (for different εj values) can be transformed into each other. I...

During and after microinjection of artificial endolymph into scala media of the guinea pig, the 2f1- f2 -DPOAE at 4.5 kHz generated by low-level primaries was recorded. Reproducible changes were measured when 1.1 microl of artificial endolymph was injected at a rate of 1.65 nl/s (1.53-1.83). This volume corresponds with an acute endolymphatic hydrops of 23%. After the onset of injection the inn...

BACKGROUND The potential carcinogenicity of naphthalene (NA), a ubiquitous environmental pollutant, in human respiratory tract is a subject of intense debate. Chief among the uncertainties in risk assessment for NA is whether human lung CYP2A13 and CYP2F1 can mediate NA's respiratory tract toxicity. OBJECTIVES We aimed to assess the in vivo function of CYP2A13 and CYP2F1 in NA bioactivation a...