In this paper, by using the q-exponential operator technique on the q-integral form of the Sears transformation formula and a Gasper q-integral formula, we obtain their generalizations. 1 Notation In this paper, we follow the notation and terminology in ([4]). For a real or complex number q (|q| < 1). let (λ)∞ = (λ; q)∞ = ∞ ∏ n=1 (1 − aq); (1.1) and let (λ : q)μ be defined by (λ)μ = (λ; q)μ = (...

In [6] it was shown that if f ∈ is starlike of order α, α= 0.294, . . . , so is the Libera integral operator F . We also know that (see, e.g., [1]) there are functions which are univalent or spiral-like in so that their Libera integral operators are not univalent or spiral-like in . Li and Owa [5] proved that if f ∈ is univalent in , then Fn(z) is starlike in |z| < 3/8. The number 3/8 is sharp....

We give elementary derivations of several classical and some new summation and transformation formulae for bilateral basic hypergeometric series. For purpose of motivation, we review our previous simple proof (“A simple proof of Bailey’s very-well-poised 6ψ6 summation”, Proc. Amer. Math. Soc., to appear) of Bailey’s very-well-poised 6ψ6 summation. Using a similar but different method, we now gi...

It is suggested that the (p, q)-hypergeometric series studied by Burban and Klimyk (in Integral Transforms and Special Functions 2 (1994) 15-36) can be considered as a special case of a more general (P, Q)-hypergeometric series. 1. Introduction I propose a general (P, Q)-hypergeometric series. In this, I am inspired mainly by the paper of Burban and Klimyk [1] titled " P, Q-differentiation, P, ...

Youth who switch schools are more likely to demonstrate a wide array of negative behavioral and educational outcomes, including dropping out of high school. However, whether switching schools actually puts youth at risk for dropout is uncertain, since youth who switch schools are similar to dropouts in their levels of prior school achievement and engagement, which suggests that switching school...

Richard A. (Dick) Askey1 was born June 4, 1933 in St. Louis, Missouri. He received his PhD at Princeton University in 1961 under the direction of Salomon Bochner. After instructorships at Washington University and the University of Chicago he joined the faculty of the University of Wisconsin-Madison in 1963, where he became full professor in 1968. Since 2003 he is Professor Emeritus at that sam...

In this talk I will present the conjecture and the main ideas that led to the proof. The proof of Lenard Weinstein (1991) [4] follows, and I will show how the two proofs are interrelated. Both proofs depend on polynomial systems that are directly related with the Koebe function. At this point algorithms of computer algebra come into the play, and I will give a computer demonstration how importa...

The Bieberbach conjecture about the coefficients of univalent functions of the unit disk was formulated by Ludwig Bieberbach in 1916 [4]. The conjecture states that the coefficients of univalent functions are majorized by those of the Koebe function which maps the unit disk onto a radially slit plane. The Bieberbach conjecture was quite a difficult problem, and it was surprisingly proved by Lou...

Gaspers installed in commercial airliner cabins are used to improve passengers' thermal comfort. To understand the impact of gasper airflow on the air quality in a cabin, this investigation measured the distributions of air velocity, air temperature, and gaseous contaminant concentration in five rows of the economy-class section of an MD-82 commercial aircraft. The gaseous contaminant was simul...

for all nonnegative integers n (Problem E3376 of Amer. Math. Monthly 97, March 1990, proposed by R. J. Blodgelt). Up to now not too much is known concerning structure and symbolic manipulation of binomial multiple-sum identities. Thus the interest in a ”computer-generated” proof of (1.1) or, more general, of identities of similar type basically arises from the question whether similar algorithm...