TY - JOUR
T1 - Totally positive kernels, pólya frequency functions, and generalized hypergeometric series
AU - Richards, Donald St P.
N1 - Funding Information:
*This work was supported in part by National Science Foundation by the Center for Advanced Studies, University of Virginia.
PY - 1990
Y1 - 1990
N2 - Recently, K. I. Gross and the author [J. Approx. Theory 59:224-246 (1989)] analyzed the total positivity of kernels of the form K(x,y) = pFq(xy), x,y ε{lunate} R, wherepFq denotes a classical generalized hypergeometric series. There, we related the determinants which define the total positivity of K to the hypergeometric functions of matrix argument, defined on the space of n × n Hermitian matrices. In the first part of this paper, we apply these methods to derive the total positivity properties of the kernels K for more general choices of the parameters in these hypergeometric series. In particular, we prove that if ai > 0 and ki is a positive integer (i = 1,...,p) then K(x,y) = pFq(a1,...,ap;a1 + k1,...,ap + kp; xy) is strictly totally positive on R2. In the second part, we use the theory of entire functions to derive some Pólya frequency function properties of the hypergeometric series pFq(x). Some of these latter results suggest a curious duality with the total positivity properties described above. For instance, we prove that if p > 1 and the ai and ki are as above, then there exists a probability density function f on R, such that f is a strict Pólya frequency function, and 1/pFp(a1+k1,...,ap +kp;a1,...,ap;z=Lf(z), the Laplace transform of f.
AB - Recently, K. I. Gross and the author [J. Approx. Theory 59:224-246 (1989)] analyzed the total positivity of kernels of the form K(x,y) = pFq(xy), x,y ε{lunate} R, wherepFq denotes a classical generalized hypergeometric series. There, we related the determinants which define the total positivity of K to the hypergeometric functions of matrix argument, defined on the space of n × n Hermitian matrices. In the first part of this paper, we apply these methods to derive the total positivity properties of the kernels K for more general choices of the parameters in these hypergeometric series. In particular, we prove that if ai > 0 and ki is a positive integer (i = 1,...,p) then K(x,y) = pFq(a1,...,ap;a1 + k1,...,ap + kp; xy) is strictly totally positive on R2. In the second part, we use the theory of entire functions to derive some Pólya frequency function properties of the hypergeometric series pFq(x). Some of these latter results suggest a curious duality with the total positivity properties described above. For instance, we prove that if p > 1 and the ai and ki are as above, then there exists a probability density function f on R, such that f is a strict Pólya frequency function, and 1/pFp(a1+k1,...,ap +kp;a1,...,ap;z=Lf(z), the Laplace transform of f.
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U2 - 10.1016/0024-3795(90)90139-4
DO - 10.1016/0024-3795(90)90139-4
M3 - Article
AN - SCOPUS:34247369226
SN - 0024-3795
VL - 137-138
SP - 467
EP - 478
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
IS - C
ER -