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{\bf
My So far unrealized Dream of Having Mourad Number One
}
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{\it  Doron ZEILBERGER}

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{\bf Abstract}: Even though Mourad Ismail is only six years my elder, I always considered him as a mentor.

Way way back, in the early eighties, I came across Mourad's very interesting papers with Dick Askey, Thaana Rashed,
and M.V. Tamahankar. In particular, in [GRZ], together with Joe Gillis and Bruce Reznick, we observed that
the fact that the power-series expansion of, for $0 \leq \lambda  \leq 1$,
$$
(1- (1-\lambda)x-\lambda y- \lambda xz - (1-\lambda)yz+xyz)^{-\beta} \quad,
$$
all have positive power series coefficients when $\beta=1$ brilliantly proved in [IT] via MacMahon's Master Theorem (one of my favorite theorems!), in a purely
elementary and combinatorial way, immediately implies Tom Koornwinder's result [K] that it does so for all $\beta \geq 1$.
Koornwider's proof was far from elementary and was rather complicated.

I wrote Mourad a {\it fan mail} letter, and expressed the wish to collaborate, so that I will have
{\it collaboration distance one} with him. I also added that this may help bring peace between
Israel and Egypt (at the time they were `enemies').

He replied with a very nice letter, that he will be happy to.

When I was up for tenure, in 1986, at Drexel University, after being hired in 1983 as a tenure-track associate professor,
he was one of the people asked to write letters. His letter was so {\bf positive} urging them
that I was more than ready also to be promoted to {\it full professor}.

While I don't know it for sure, he probably wrote me letters when I moved from Drexel to Temple in 1990.

While my dream to collaborate with Mourad never came to be, I lived it {\it vicariously}. My  mentor
Joe Gillis, together with at-the-time his grad student, T. Offer, wrote a beautiful paper [GIO].
So my collaboration distance with Mourad is {\bf two} via numerous routes.

The paper[GIO] is about the asymptotics of the number of multi-set derangements, extending the work in [AIR].
It is asymptotic analysis at its best.
To my great surprise, and disappointment, I could not find it in {\it MathSciNet},
someone messed up! There are probably numerous other gems that MathSciNet (formerly Mathematics Reviews) missed.


Finally, one of my {\bf all time favorites} is Mourad's {\it A simple proof of Ramanujan's ${}_1 \psi_1$ sum}, that
was published in 1977 in the Proceedings of the AMS, and was immortalized in George Andrews' classic
`Ten Lectures on $q$-series'' {\bf CBMS} 1986 (based on Andrews' lectures in a 1985 conference co-organized by
Mourad at Arizona State University).

Thanks, Nourad, for all that great math, and warm friendship and mentoring.

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{\bf References}

[AsIs] R. Askey and M.E.H. Ismail, {\it Permutation problems and special functions}, Canad. J. Math. {\bf 28} (1976), 853-874.

[AIK] R. Askey, M.E.H. Ismail, and T. Koornwinder, {\it Weighted permutation problems and Laguerre polynomials},
J. Comb. Theory (Ser. A.) {\bf 25} (1978), 277-287.

[AIR] R. Askey, M.E.H. Ismail, and M.T. Rashed, {\it A derangement problem}. 
Technical report 1522, UW-Madison Mathematics Research Center, June 1975. 

[G] J. Gillis, {\it The statistics of derangement-a survey}, Journal of Statistical Physics {\bf 58} (1990),575-578.

[GIO]  J. Gillis, M. E. H. Ismail, and T. Offer, {\it An Asymptotic Problem in Derangement Theory},
SIAM Journal on Mathematical Analysis {\bf 21} (1990), 262-269.

[GRZ] J. Gillis, B. Reznick,  and D. Zeilberger {\it On Elementary methods in positivity theory},
SIAM J. of Mathematical Analysis {\bf 14} (1983), 396-399.

[IT]  M.E.H. Ismail and M.V. Tamahankar, {\it A combinatorial approach to some positivity problems}, SIAM J. Math. Anal. {\bf 10} (1979), 478-485.

[K]  T. Koornwinder, {\it Positivity proofs for linearization and connection coefficients of orthogonal polynomials},
J. London Math. Soc. {\bf 2} (1978), 101-114.

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Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen
Rd., Piscataway, NJ 08854-8019, USA. \hfill\break
Email: {\tt DoronZeil at gmail  dot com}   \quad .

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{\bf Sept. 16, 2026} 

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