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21: 28.20 Definitions and Basic Properties

… ►Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to ${\zeta }^{1/2}{\mathrm{e}}^{\pm 2\mathrm{i}h\zeta }$ as $\zeta \to \mathrm{\infty }$ in the respective sectors $|\mathrm{ph}\left(\mp \mathrm{i}\zeta \right)|\le \frac{3}{2}\pi -\delta$, $\delta$ being an arbitrary small positive constant. It follows that (28.20.1) has independent and unique solutions ${\mathrm{M}}_{\nu }^{(3)}(z,h)$, ${\mathrm{M}}_{\nu }^{(4)}(z,h)$ such that …In addition, there are unique solutions ${\mathrm{M}}_{\nu }^{(1)}(z,h)$, ${\mathrm{M}}_{\nu }^{(2)}(z,h)$ that are real when $z$ is real and have the properties …

22: 1.14 Integral Transforms

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Uniqueness

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Uniqueness

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23: 32.11 Asymptotic Approximations for Real Variables

… ►Conversely, for any nonzero real $k$, there is a unique solution $w_k(x)$ of (32.11.4) that is asymptotic to $k\mathrm{Ai}\left(x\right)$ as $x\to +\mathrm{\infty }$. … ►Conversely, for any $h$ $(\ne 0)$ there is a unique solution $w_h(x)$ of (32.11.29) that is asymptotic to $h{U}^2(-\nu -\frac{1}{2},\sqrt{2}x)$ as $x\to +\mathrm{\infty }$. …

24: 3.1 Arithmetics and Error Measures

… ►with $x\ge 0$ and $\mathrm{\ell }$ the unique nonnegative integer such that $a\equiv {\ln }_{\mathrm{\ell }}(x)\in [0,1)$. …

25: 18.27 $q$-Hahn Class

… ►Some of the systems of OP’s that occur in the classification do not have a unique orthogonality property. … ►The measure is not uniquely determined: … ►The measure is not uniquely determined: … ►For discrete $q$-Hermite II polynomials the measure is not uniquely determined. …

26: 18.28 Askey–Wilson Class

… ►If, in addition to (18.28.11) or (18.28.12), we have $a^{-1}b\le q$, then the measure in (18.28.10) is the unique orthogonality measure. … ►For continuous $q^{-1}$-Hermite polynomials the orthogonality measure is not unique. …

27: 18.37 Classical OP’s in Two or More Variables

… ►The following three conditions, taken together, determine $R_{m,n}^{(\alpha )}\left(z\right)$ uniquely: …

28: 24.16 Generalizations

… ►Let ${\chi }_0$ be the trivial character and ${\chi }_4$ the unique (nontrivial) character with $f=4$; that is, ${\chi }_4(1)=1$, ${\chi }_4(3)=-1$, ${\chi }_4(2)={\chi }_4(4)=0$. …

29: 27.2 Functions

… ►Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer $n>1$ can be represented uniquely as a product of prime powers, …

30: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

… ►The functions $f_n(z,q)$, $g_n(z,q)$ are unique. …