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With the help of El Bachraoui's lemma, the creative microscoping method, and a new form of the Chinese remainder theorem for coprime polynomials, we prove a $q$-supercongruence for double series and a $q$-supercongruence for triple series modulo the sixth power of a cyclotomic polynomial. As conclusions, two corresponding supercongruences for double and triple series, which are associated with the (D.2) supercongruence of Van Hamme, are given.

Harmonic numbers are significant in various branches of number theory. With the help of the digamma function, we prove ten conjectural series of Z.-W. Sun involving harmonic numbers. Several ones of them are also series expansions of $\log2/\pi^2$.

Series expansions for the mathematical constant $\beta(4)$ are rare in the history. With the help of the operator method and a hypergeometric transformation, we prove a surprising conjectural series of Sun for $\beta(4)$. Furthermore, we find five new series for the same constant in this paper.

In terms of the operator method, we prove two conjectural series for $\pi$ of Sun involving harmonic numbers of order two. Furthermore, we also give $q$-analogues of six $\pi$-formulas including the two ones just mentioned.

With the help of a summation of basic hypergeometric series, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials, we find some new $q$-supercongruences. Especially, we give a $q$-analogue of a formula due to Liu [J. Math. Anal. Appl. 497 (2021), Art.~124915].

$q$-Supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial are very rare in the literature. In this paper, we establish some $q$-supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial in terms of Jackson's $_8\phi_7$ summation, Watson's $_8\phi_7$ transformation, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials. More concretely, we give a $q$-analogue of a nice formula due to Long and Ramakrishna [Adv. Math. 290 (2016), 773--808] and two $q$-supercongruences involving double series.

In terms of the creative microscoping method recently introduced by Guo and Zudilin and the Chinese remainder theorem for coprime polynomials, we establish a $q$-supercongruence with two parameters modulo $[n]\Phi_n(q)^3$. Here $[n]=(1-q^n)/(1-q)$ and $\Phi_n(q)$ is the $n$-th cyclotomic polynomial in $q$. In particular, we confirm a recent conjecture of Guo and give a complete $q$-analogue of Long's supercongruence. The latter is also a generalization of a recent $q$-supercongruence obtained by Guo and Schlosser.

According to the $q$-series method, a short proof for Hou and Sun's identity, which is the $q$-analogue of a known $\pi$-formula, is offered. Furthermore, $q$-analogues of several other $\pi$-formulas are also established in terms of the $q$-series method and the telescoping method.

By means of the derivative operator and Chu-Vandermonde convolution, four families of summation formulas involving harmonic numbers with even or odd indexes are established.

By means of the derivative operator and Whipple-type $_3F_2$-series identities, two families of summation formulae involving generalized harmonic numbers are established., Comment: arXiv admin note: substantial text overlap with arXiv:1607.01006, arXiv:1606.09496

In terms of the derivative operator and Watson-type $_3F_2$-series identities, three families of summation formulae involving generalized harmonic numbers are established., Comment: arXiv admin note: text overlap with arXiv:1606.09496

In terms of the derivative operator, integral operator and Saalsch\"{u}tz's theorem, two families of summation formulae involving generalized harmonic numbers are established.

In terms of the derivative operator and three hypergeometric series identities, several interesting summation formulas involving generalized harmonic numbers are established.

Solutions of Pell's equation and hypergeometric series identities are used to study series expansions for $\sqrt{p}$ where $p$ are arbitrary prime numbers. Numerous fast convergent series expansions for this family of irrational numbers are established.

Combining the derivative operator with a binomial sum from the telescoping method, we establish a family of summation formulas involving generalized harmonic numbers.

Combining the derivative operator with Chu-Vandermonde convolution, we establish a class of summation formulas on generalized harmonic numbers., Comment: This paper has been withdrawn by the author because that the main results of it are trivial

In terms of the telescoping method, a simple binomial sum is given. By applying the derivative operators to the equation just mentioned, we establish several general harmonic number identities including some known results.

In terms of the hypergeometric method, we establish the extensions of two formulas for $1/\pi$ due to Ramanujan [27]. Further, other five summation formulas for $1/\pi$ with free parameters are also derived in the same way.