Ramanujan 1ψ1 summation

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1: 17.18 Methods of Computation

… ►The two main methods for computing basic hypergeometric functions are: (1) numerical summation of the defining series given in §§17.4(i) and 17.4(ii); (2) modular transformations. Method (1) is applicable within the circles of convergence of the defining series, although it is often cumbersome owing to slowness of convergence and/or severe cancellation. …Lehner (1941) uses Method (2) in connection with the Rogers–Ramanujan identities. ►Method (1) can sometimes be improved by application of convergence acceleration procedures; see §3.9. …

2: 27.14 Unrestricted Partitions

… ►

§27.14(v) Divisibility Properties

►Ramanujan (1921) gives identities that imply divisibility properties of the partition function. For example, the Ramanujan identity … ►

§27.14(vi) Ramanujan’s Tau Function

… ►

3: 27.10 Periodic Number-Theoretic Functions

… ►An example is Ramanujan’s sum: …where

\chi_{1}

is the principal character (mod

k

). … ►Another generalization of Ramanujan’s sum is the Gauss sum

G\left(n,\chi\right)

associated with a Dirichlet character

\chi\pmod{k}

. …In particular,

G\left(n,\chi_{1}\right)=c_{k}\left(n\right)

. … ►For any Dirichlet character

\chi\pmod{k}

G\left(n,\chi\right)

is separable for

n

\left(n,k\right)=1

, and is separable for every

n

if and only if

G\left(n,\chi\right)=0

whenever

\left(n,k\right)>1

. …

4: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

… ►

Ramanujan’s ${{}_{1}\psi_{1}}$ Summation

►

17.8.2

{{}_{1}\psi_{1}}\left({a\atop b};q,z\right)=\frac{\left(q,b/a,az,q/(az);q% \right)_{\infty}}{\left(b,q/a,z,b/(az);q\right)_{\infty}}.

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.8.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §17.8, §17.8 and Ch.17

… ►

17.8.3

\sum_{n=-\infty}^{\infty}(-1)^{n}q^{n(3n-1)/2}z^{3n}(1+zq^{n})=\left(q,-z,-q/z% ;q\right)_{\infty}\left(qz^{2},q/{z^{2}};q^{2}\right)_{\infty}.

ⓘ

Symbols:: $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §17.8, §17.8 and Ch.17

►

Bailey’s Bilateral Summations

… ►

17.8.6

{{}_{4}\psi_{4}}\left({-qa^{\frac{1}{2}},b,c,d\atop-a^{\frac{1}{2}},aq/b,aq/c,% aq/d};q,\frac{qa^{\frac{3}{2}}}{bcd}\right)=\frac{\left(aq,aq/(bc),aq/(bd),aq/% (cd),qa^{\frac{1}{2}}/b,qa^{\frac{1}{2}}/c,qa^{\frac{1}{2}}/d,q,q/a;q\right)_{% \infty}}{\left(aq/b,aq/c,aq/d,q/b,q/c,q/d,qa^{\frac{1}{2}},qa^{-\frac{1}{2}},% qa^{\frac{3}{2}}/(bcd);q\right)_{\infty}},

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.8.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §17.8, §17.8 and Ch.17

…

5: 20.11 Generalizations and Analogs

… ►

§20.11(ii) Ramanujan’s Theta Function and $q$ -Series

►Ramanujan’s theta function

f(a,b)

is defined by … ►

§20.11(iii) Ramanujan’s Change of Base

… ►Similar identities can be constructed for

{{}_{2}F_{1}}\left(\tfrac{1}{3},\tfrac{2}{3};1;k^{2}\right)

{{}_{2}F_{1}}\left(\tfrac{1}{4},\tfrac{3}{4};1;k^{2}\right)

, and

{{}_{2}F_{1}}\left(\tfrac{1}{6},\tfrac{5}{6};1;k^{2}\right)

. These results are called Ramanujan’s changes of base. …

6: George E. Andrews

… ►Ramanujan, whose last notebook he unearthed in 1976. … ►Andrews served as President of the AMS from February 1, 2009 to January 31, 2011, and became a Fellow of the AMS in 2012. …

7: 27.20 Methods of Computation: Other Number-Theoretic Functions

… ►To compute a particular value

p\left(n\right)

it is better to use the Hardy–Ramanujan–Rademacher series (27.14.9). … ►A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function

\tau\left(n\right)

, and the values can be checked by the congruence (27.14.20). …

8: 20.12 Mathematical Applications

… ►For applications of Jacobi’s triple product (20.5.9) to Ramanujan’s

\tau\left(n\right)

function and Euler’s pentagonal numbers see Hardy and Wright (1979, pp. 132–160) and McKean and Moll (1999, pp. 143–145). … ►The space of complex tori

\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})

(that is, the set of complex numbers

z

in which two of these numbers

z_{1}

and

z_{2}

are regarded as equivalent if there exist integers

m, n

such that

z_{1}-z_{2}=m+\tau n

) is mapped into the projective space

P^{3}

via the identification

z\to(\theta_{1}\left(2z\middle|\tau\right),\theta_{2}\left(2z\middle|\tau% \right),\theta_{3}\left(2z\middle|\tau\right),\theta_{4}\left(2z\middle|\tau% \right))

. …

9: 27.21 Tables

… ►9 lists all primes that are less than 1 00000. … ►Tables of the Ramanujan function

\tau\left(n\right)

are published in Lehmer (1943) and Watson (1949). …

10: Frank Garvan

… ►He is managing editor of the Ramanujan Journal, a journal devoted to areas of mathematics influenced by Ramanujan. …