ムーラッド・イスマイル

**ムーラッド・イスマイル** (Mourad Ismail, 1944年4月27日 - ) は、エジプト出身の数学者である。専門は特殊関数論と直交多項式論である。

## 主な業績
イスマイルは、特殊関数と直交多項式に関する研究において数々の重要な貢献を果たしている。彼の名を冠した多項式としては、以下のものが挙げられる。

* ロジャース-アスキー-イスマイル多項式 (Rogers-Askey-Ismail polynomials)
* アルサラム-イスマイル多項式 (Al-Salam-Ismail polynomials)
* チハラ-イスマイル多項式 (Chihara-Ismail polynomials)

## 主な共同研究者
長年にわたり、以下の著名な数学者たちと共同研究を行っている。

* リチャード・アスキー (Richard Askey)
* ワリード・アルサラム (Waleed Al-Salam)
* ミザン・ラーマン (Mizan Rahman)

## 著書・解説記事

* Askey, R., & Ismail, M. (1984). *Recurrence relations, continued fractions and orthogonal polynomials*. American Mathematical Society.
* Ismail, Mourad E. H. (2005). *Classical and Quantum Orthogonal Polynomials in One Variable*. Cambridge University Press. 
* Ismail, M. E., & Zhang, R. (2017). *A review of multivariate orthogonal polynomials*. Journal of the Egyptian Mathematical Society, 25(2), 91-110.
* Ismail, Mourad E. H. (編著) (2020). *Encyclopedia of Special Functions: The Askey–Bateman Project, Volume 1: Univariate Orthogonal Polynomials*. Cambridge University Press.

## 代表的な論文
イスマイルは、多くの学術論文を発表しており、その一部を以下に示す。

### 共著
**1980年代**

* Ismail, M. E., & Wilson, J. A. (1982). Asymptotic and generating relations for the q-Jacobi and 4φ3 polynomials. *Journal of Approximation Theory*, 36(1), 43-54.
* Ismail, M. E., Lorch, L., & Muldoon, M. E. (1986). Completely monotonic functions associated with the gamma function and its q-analogues. *Journal of Mathematical Analysis and Applications*, 116(1), 1-9.
* Bustoz, J., & Ismail, M. E. (1986). On gamma function inequalities. *Mathematics of Computation*, 47(176), 659-667.
* Ismail, M. E., Stanton, D., & Viennot, G. (1987). The combinatorics of q-Hermite polynomials and the Askey—Wilson integral. *European Journal of Combinatorics*, 8(4), 379-392.
* Ismail, M. E., & Muldoon, M. E. (1988). On the variation with respect to a parameter of zeros of Bessel and q-Bessel functions. *Journal of Mathematical Analysis and Applications*, 135(1), 187-207.
* Ismail, M. E., & Stanton, D. (1988). On the Askey-Wilson and Rogers polynomials. *Canadian Journal of Mathematics*, 40(5), 1025-1045.

**1990年代**

* Ismail, M. E., & Rahman, M. (1991). The associated Askey-Wilson polynomials. *Transactions of the American Mathematical Society*, 328(1), 201-237.
* Ismail, M. E., Masson, D. R., & Rahman, M. (1991). Complex weight functions for classical orthogonal polynomials. *Canadian Journal of Mathematics*, 43(6), 1294-1308.
* Ismail, Mourad EH, & Ruedemann, Richard W. (1992). Relation between polynomials orthogonal on the unit circle with respect to different weights. *Journal of Approximation Theory*, 71(1), 39-60.
* Ismail, M. E., & Masson, D. R. (1994). q-Hermite polynomials, biorthogonal rational functions, and q-beta integrals. *Transactions of the American Mathematical Society*, 346(1), 63-116.
* Ismail, M., & Rahman, M. (1995). Some basic bilateral sums and integrals. *Pacific Journal of Mathematics*, 170(2), 497-515.
* Ismail, M. E., & Masson, D. R. (1995). Generalized orthogonality and continued fractions. *Journal of Approximation Theory*, 83(1), 1-40.
* Brown, B. M., & Ismail, M. E. (1995). A right inverse of the Askey-Wilson operator. *Proceedings of the American Mathematical Society*, 123(7), 2071-2079.
* Chen, Y., & Ismail, M. E. (1997). Ladder operators and differential equations for orthogonal polynomials. *Journal of Physics A: Mathematical and General*, 30(22), 7817.
* Ismail, M. E., & Stanton, D. (1997). Classical orthogonal polynomials as moments. *Canadian Journal of Mathematics*, 49(3), 520-542.
* Ismail, M. E., & Rahman, M. (1998). The q-Laguerre Polynomials and Related Moment Problems. *Journal of Mathematical Analysis and Applications*, 218(1), 155-174.
* Ismail, M., Rahman, M., & Stanton, D. (1999). Quadratic q-exponentials and connection coefficient problems. *Proceedings of the American Mathematical Society*, 127(10), 2931-2941.

**2000年代**

* Ismail, M. E., & Pitman, J. (2000). Algebraic evaluations of some Euler integrals, duplication formulae for Appell's hypergeometric function F1, and Brownian variations. *Canadian Journal of Mathematics*, 52(5), 961-981.
* Ismail, M. E., & Stanton, D. W. (2000). Addition theorems for the q-exponential function. *Contemporary Mathematics*, 254, 235-246.
* Ismail, M. E., Valent, G., & Yoon, G. J. (2001). Some orthogonal polynomials related to elliptic functions. *Journal of Approximation Theory*, 112(2), 251-278.
* Ismail, M. E., & Stanton, D. (2002). q-Integral and moment representations for q-orthogonal polynomials. *Canadian Journal of Mathematics*, 54(4), 709-735.
* Ismail, M. E., & Stanton, D. (2003). Applications of q-Taylor theorems. *Journal of Computational and Applied Mathematics*, 153(1-2), 259-272.

**2010年代**

* Ismail, M., & Simeonov, P. (2015). Complex Hermite polynomials: their combinatorics and integral operators. *Proceedings of the American Mathematical Society*, 143(4), 1397-1410.
* Genest, V., Ismail, M., Vinet, L., & Zhedanov, A. (2016). Tridiagonalization of the hypergeometric operator and the Racah–Wilson algebra. *Proceedings of the American Mathematical Society*, 144(10), 4441-4454.
* Ismail, M. E. H., & Zhang, R. (2016). Classes of bivariate orthogonal polynomials. *Symmetry, Integrability and Geometry: Methods and Applications*, 12(0), 21-37.
* Ismail, M., & Zhang, R. (2017). On some 2D orthogonal q-polynomials. *Transactions of the American Mathematical Society*, 369(10), 6779-6821.
* Ismail, M., & Zhang, R. (2018). q-Bessel functions and Rogers-Ramanujan type identities. *Proceedings of the American Mathematical Society*, 146(9), 3633-3646.

### 単著

* Ismail, M. E. (1977). A simple proof of Ramanujan's 1ψ1 sum. *Proceedings of the American Mathematical Society*, 63(1), 185-186.
* Ismail, M. E. (1981). The basic Bessel functions and polynomials. *SIAM Journal on Mathematical Analysis*, 12(3), 454-468.
* Ismail, M. E. (1982). *Journal of Mathematical Analysis and Applications*, 86(1), 1-19.
* Ismail, M. (2016). Analytic properties of complex Hermite polynomials. *Transactions of the American Mathematical Society*, 368(2), 1189-1210.

## 外部リンク
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