# Bochner theorem

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## 1—10 of 122 matching pages

Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …
##### 3: 18.3 Definitions
• 1.

As eigenfunctions of second order differential operators (Bochner’s theorem, Bochner (1929)). See the differential equations $A(x)p_{n}^{\prime\prime}(x)+B(x)p_{n}^{\prime}(x)+\lambda_{n}p_{n}(x)=0$, in Table 18.8.1.

• ##### 4: 27.15 Chinese Remainder Theorem
###### §27.15 Chinese Remainder Theorem
The Chinese remainder theorem states that a system of congruences $x\equiv a_{1}\pmod{m_{1}},\dots,x\equiv a_{k}\pmod{m_{k}}$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli. This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod $m_{1}$), (mod $m_{2}$), (mod $m_{3}$), and (mod $m_{4}$), respectively. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $\pmod{m}$, which is correct to 20 digits. …
##### 5: Bibliography B
• H. F. Baker (1995) Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions. Cambridge University Press, Cambridge.
• M. V. Berry (1976) Waves and Thom’s theorem. Advances in Physics 25 (1), pp. 1–26.
• S. Bochner (1929) Über Sturm-Liouvillesche Polynomsysteme. Math. Z. 29 (1), pp. 730–736.
• S. Bochner (1952) Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.
• S. Bochner and W. T. Martin (1948) Several Complex Variables. Princeton Mathematical Series, Vol. 10, Princeton University Press, Princeton, N.J..
• ##### 6: Bibliography K
• Y. S. Kim, A. K. Rathie, and R. B. Paris (2013) An extension of Saalschütz’s summation theorem for the series ${}_{r+3}F_{r+2}$ . Integral Transforms Spec. Funct. 24 (11), pp. 916–921.
• B. J. King and A. L. Van Buren (1973) A general addition theorem for spheroidal wave functions. SIAM J. Math. Anal. 4 (1), pp. 149–160.
• T. H. Koornwinder (1975a) A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. 13, pp. 145–159.
• K. H. Kwon, L. L. Littlejohn, and G. J. Yoon (2006) Construction of differential operators having Bochner-Krall orthogonal polynomials as eigenfunctions. J. Math. Anal. Appl. 324 (1), pp. 285–303.
• ##### 7: 30.10 Series and Integrals
For an addition theorem, see Meixner and Schäfke (1954, p. 300) and King and Van Buren (1973). …
##### 8: 10.44 Sums
###### §10.44(i) Multiplication Theorem
Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute $\pi$ to high precision (Borwein and Borwein (1987, p. 26)). …