values at q=0

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21: 31.7 Relations to Other Functions

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31.7.1 $${}_2{}^{}F_1^{}(\alpha ,\beta ;\gamma ;z)=H\mathrm{\ell }(1,\alpha \beta ;\alpha ,\beta ,\gamma ,\delta ;z)=H\mathrm{\ell }(0,0;\alpha ,\beta ,\gamma ,\alpha +\beta +1-\gamma ;z)=H\mathrm{\ell }(a,a\alpha \beta ;\alpha ,\beta ,\gamma ,\alpha +\beta +1-\gamma ;z).$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $a$: complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

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http://dlmf.nist.gov/31.7.E1

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Annotations for §31.7(i), §31.7 and Ch.31

►Other reductions of $H\mathrm{\ell }$ to a ${}_2{}^{}F_1^{}$, with at least one free parameter, exist iff the pair $(a,p)$ takes one of a finite number of values, where $q=\alpha \beta p$. … ►With $z={\mathrm{sn}}^2(\zeta ,k)$ and … ►

$q=-\frac{1}{4}ah,$

… ►Similar specializations of formulas in §31.3(ii) yield solutions in the neighborhoods of the singularities $\zeta =K$, $K+\mathrm{i}{K}^{\prime }$, and $\mathrm{i}{K}^{\prime }$, where $K$ and $K^{\prime }$ are related to $k$ as in §19.2(ii).

22: 14.16 Zeros

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$\mu \le 0$.

… ► ${𝖰}_{\nu }^{\mu }\left(x\right)$ has $\max (\lceil \nu -|\mu |\rceil ,0)+k$ zeros in the interval $(-1,1)$, where $k$ can take one of the values $-1$, $0$, $1$, $2$, subject to $\max (\lceil \nu -|\mu |\rceil ,0)+k$ being even or odd according as $\cos \left(\nu \pi \right)$ and $\cos \left(\mu \pi \right)$ have opposite signs or the same sign. In the special case $\mu =0$ and $\nu =n=0,1,2,3,\mathrm{\dots }$, ${𝖰}_n\left(x\right)$ has $n+1$ zeros in the interval . … ►For all other values of $\mu$ and $\nu$ (with $\nu \ge -\frac{1}{2}$) $P_{\nu }^{\mu }\left(x\right)$ has no zeros in the interval $(1,\mathrm{\infty })$. ► ${𝑸}_{\nu }^{\mu }\left(x\right)$ has no zeros in the interval $(1,\mathrm{\infty })$ when $\nu >-1$, and at most one zero in the interval $(1,\mathrm{\infty })$ when .

23: 3.8 Nonlinear Equations

… ►From this graph we estimate an initial value $x_0=4.65$. …The convergence is faster when we use instead the function $f(x)=x\cos x-\sin x$; in addition, the successful interval for the starting value $x_0$ is larger. … ►We construct sequences $q_j$ and $r_j$, $j=n+1,n,\mathrm{\dots },0$, from $q_{n+1}=r_{n+1}=0$, $q_n=r_n=a_n$, and for $j\le n-1$, … ►The method converges locally and quadratically, except when the wanted quadratic factor is a multiple factor of $q(z)$. … ►With the starting values $s_0=\frac{7}{4}$, $t_0=-\frac{1}{2}$, an approximation to the quadratic factor $z^2-2z+1={(z-1)}^2$ is computed ($s=2$, $t=-1$). …

24: 16.17 Definition

… ►Assume also that $m$ and $n$ are integers such that $0\le m\le q$ and $0\le n\le p$, and none of $a_k-b_j$ is a positive integer when $1\le k\le n$ and $1\le j\le m$. … ►

$L$ is a loop that starts at infinity on a line parallel to the positive real axis, encircles the poles of the $\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-s\right)$ once in the negative sense and returns to infinity on another line parallel to the positive real axis. The integral converges for all $z$ ($\ne 0$) if , and for if $p=q\ge 1$.

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$L$ is a loop that starts at infinity on a line parallel to the negative real axis, encircles the poles of the $\mathrm{\Gamma }\left(1-a_{\mathrm{\ell }}+s\right)$ once in the positive sense and returns to infinity on another line parallel to the negative real axis. The integral converges for all $z$ if $p>q$, and for $|z|>1$ if $p=q\ge 1$.

… ►When more than one of Cases (i), (ii), and (iii) is applicable the same value is obtained for the Meijer $G$-function. ►Assume $p\le q$, no two of the bottom parameters $b_j$, $j=1,\mathrm{\dots },m$, differ by an integer, and $a_j-b_k$ is not a positive integer when $j=1,2,\mathrm{\dots },n$ and $k=1,2,\mathrm{\dots },m$. …

25: 18.24 Hahn Class: Asymptotic Approximations

… ►When the parameters $\alpha$ and $\beta$ are fixed and the ratio $n/N=c$ is a constant in the interval (0,1), uniform asymptotic formulas (as $n\to \mathrm{\infty }$ ) of the Hahn polynomials $Q_n(z;\alpha ,\beta ,N)$ can be found in Lin and Wong (2013) for $z$ in three overlapping regions, which together cover the entire complex plane. … ►With $x=\lambda N$ and $\nu =n/N$, Li and Wong (2000) gives an asymptotic expansion for $K_n(x;p,N)$ as $n\to \mathrm{\infty }$, that holds uniformly for $\lambda$ and $\nu$ in compact subintervals of $(0,1)$. … ►Asymptotic approximations are also provided for the zeros of $K_n(x;p,N)$ in various cases depending on the values of $p$ and $\mu$. … ►The first expansion holds uniformly for $\delta \le x\le 1+\delta$, and the second for $1-\delta \le x\le 1+{\delta }^{-1}$, $\delta$ being an arbitrary small positive constant. … ►Taken together, these expansions are uniformly valid for and for $a$ in unbounded intervals—each of which contains $[0,(1-\delta )n]$, where $\delta$ again denotes an arbitrary small positive constant. …

26: 22.11 Fourier and Hyperbolic Series

… ►Throughout this section $q$ and $\zeta$ are defined as in §22.2. ►If , then … ►Next, if , then …In (22.11.7)–(22.11.12) the left-hand sides are replaced by their limiting values at the poles of the Jacobian functions. ►Next, with $E=E\left(k\right)$ denoting the complete elliptic integral of the second kind (§19.2(ii)) and , …

27: 3.11 Approximation Techniques

… ►with initial values $T_0\left(x\right)=1$, $T_1\left(x\right)=x$. … ►There exists a unique solution of this minimax problem and there are at least $k+\mathrm{\ell }+2$ values $x_j$, , such that $m_j=m$, where … ►Thus if $b_0\ne 0$, then the Maclaurin expansion of (3.11.21) agrees with (3.11.20) up to, and including, the term in $z^{p+q}$. … ►With $b_0=1$, the last $q$ equations give $b_1,\mathrm{\dots },b_q$ as the solution of a system of linear equations. … ►Given $n+1$ distinct points $x_k$ in the real interval $[a,b]$, with ($a=$)($=b$), on each subinterval $[x_k,x_{k+1}]$, $k=0,1,\mathrm{\dots },n-1$, a low-degree polynomial is defined with coefficients determined by, for example, values $f_k$ and $f_k^{\prime }$ of a function $f$ and its derivative at the nodes $x_k$ and $x_{k+1}$. …

28: 14.20 Conical (or Mehler) Functions

… ►Another real-valued solution ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ of (14.20.1) was introduced in Dunster (1991). …${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ exists except when $\mu =\frac{1}{2},\frac{3}{2},\mathrm{\dots }$ and $\tau =0$; compare §14.3(i). … ►Lastly, for the range , $P_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ is a real-valued solution of (14.20.1); in terms of $Q_{-\frac{1}{2}\pm \mathrm{i}\tau }^{\mu }\left(x\right)$ (which are complex-valued in general): … ►where the inverse trigonometric functions take their principal values. … ►with the inverse tangent taking its principal value. …

29: 8.18 Asymptotic Expansions of $I_x(a,b)$

… ►for each $n=0,1,2,\mathrm{\dots }$. … ►with limiting value … ►All of the $c_k(\eta )$ are analytic at $\eta =0$. … ►with limiting value … ►All of the $h_k(\zeta ,\mu )$ are analytic at $\zeta =\mu$ (corresponding to $x=x_0$). …

30: 26.10 Integer Partitions: Other Restrictions

… ► $p_m(𝒟,n)$ denotes the number of partitions of $n$ into at most $m$ distinct parts. … ►where the sum is over nonnegative integer values of $k$ for which $n-\frac{1}{2}(3k^2\pm k)\ge 0$. … ►where the sum is over nonnegative integer values of $k$ for which $n-(3k^2\pm k)\ge 0$. … ►where the sum is over nonnegative integer values of $m$ for which $n-\frac{1}{2}k{m}^2-m+\frac{1}{2}km\ge 0$. … ►The quantity $A_k(n)$ is real-valued. …