# elementary identities

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See §22.17.
##### 4: Bibliography M
• S. C. Milne (1985b) An elementary proof of the Macdonald identities for $A_{l}^{(1)}$ . Adv. in Math. 57 (1), pp. 34–70.
• ##### 5: 17.17 Physical Applications
In exactly solved models in statistical mechanics (Baxter (1981, 1982)) the methods and identities of §17.12 play a substantial role. … See Kassel (1995). …
##### 6: 25.5 Integral Representations
###### §25.5(i) In Terms of Elementary Functions
25.5.6 $\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\frac{1}{\Gamma\left(s\right)}% \int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x% ^{s-1}}{e^{x}}\mathrm{d}x,$ $\Re s>-1$.
25.5.14 $\omega(x)\equiv\sum_{n=1}^{\infty}e^{-n^{2}\pi x}=\frac{1}{2}\left(\theta_{3}% \left(0\middle|ix\right)-1\right).$
##### 7: 35.7 Gaussian Hypergeometric Function of Matrix Argument
35.7.2 $P^{(\gamma,\delta)}_{\nu}\left(\mathbf{T}\right)=\frac{\Gamma_{m}\left(\gamma+% \nu+\frac{1}{2}(m+1)\right)}{\Gamma_{m}\left(\gamma+\frac{1}{2}(m+1)\right)}\*% {{}_{2}F_{1}}\left({-\nu,\gamma+\delta+\nu+\frac{1}{2}(m+1)\atop\gamma+\frac{1% }{2}(m+1)};\mathbf{T}\right),$ $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$; $\gamma,\delta,\nu\in\mathbb{C}$; $\Re\left(\gamma\right)>-1$.
35.7.4 $\lim_{c\to\infty}{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{I}-c\mathbf{T}^{-1}% \right)=\left|\mathbf{T}\right|^{b}\Psi\left(b;b-a+\tfrac{1}{2}(m+1);\mathbf{T% }\right).$
35.7.7 ${{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{I}\right)=\frac{\Gamma_{m}\left(c% \right)\Gamma_{m}\left(c-a-b\right)}{\Gamma_{m}\left(c-a\right)\Gamma_{m}\left% (c-b\right)}},$ $\Re\left(c\right),\Re\left(c-a-b\right)>\frac{1}{2}(m-1)$.
35.7.11 ${{}_{2}F_{1}}\left({a,b\atop c};\mathbf{I}-\alpha^{-1}\mathbf{T}\right)$
These approximations are in terms of elementary functions. …
##### 8: Bibliography S
• L. Shen (1998) On an identity of Ramanujan based on the hypergeometric series ${}_{2}F_{1}(\frac{1}{3},\frac{2}{3};\frac{1}{2};x)$ . J. Number Theory 69 (2), pp. 125–134.
• R. Sitaramachandrarao and B. Davis (1986) Some identities involving the Riemann zeta function. II. Indian J. Pure Appl. Math. 17 (10), pp. 1175–1186.
• SLATEC (free Fortran library)
• D. M. Smith (1989) Efficient multiple-precision evaluation of elementary functions. Math. Comp. 52 (185), pp. 131–134.
• ##### 9: Bibliography R
• J. Riordan (1979) Combinatorial Identities. Robert E. Krieger Publishing Co., Huntington, NY.
• M. D. Rogers (2005) Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.
• K. H. Rosen (2004) Elementary Number Theory and its Applications. 5th edition, Addison-Wesley, Reading, MA.
• ##### 10: 1.18 Linear 2nd Order Differential Operators and Eigenfunction Expansions
###### §1.18(v) A Continuous Spectrum of Eigenvalues: Normalization, Resolution of the Identity, and Spectral Expansions
See Friedman (1990, pages 233 – 252) for elementary discussions of both equations; also the references in §1.18(vii).