On chemical and mathematical characteristics of generalized degree–based molecular descriptors

1.   Introduction and our main results

The classical rational Ramanujan-type series for $\pi^{-1}$ (cf. [1,2,8,27] and a nice introduction by S. Cooper [10,Chapter 14]) have the form

where $b,c,m$ are integers with $bm\not = 0$, $d$ is a positive squarefree number, $\lambda$ is a nonzero rational number, and $a(k)$ is one of the products

In 1997 Van Hamme [47] conjectured that such a series $(*)$ has a $p$-adic analogue of the form

where $p$ is any odd prime with $p\nmid dm$ and $\lambda\in{\Bbb Z}_p$, ${\varepsilon}_1\in\{\pm1\}$ and ${\varepsilon}_d = -1$ if $d>1$. (As usual, ${\Bbb Z}_p$ denotes the ring of all $p$-adic integers, and $( \frac{\cdot}p)$ stands for the Legendre symbol.) W. Zudilin [53] followed Van Hamme's idea to provide more concrete examples. Sun [33] realized that many Ramanujan-type congruences are related to Bernoulli numbers or Euler numbers. In 2016 the author [44] thought that all classical Ramanujan-type congruences have their extensions like

where $p$ is an odd prime, and $n\in{\Bbb Z}^+ = \{1,2,3,\ldots\}$. See Sun [45,Conjectures 21-24] for more such examples and further refinements involving Bernoulli or Euler numbers.

During the period 2002–2010, some new Ramanujan-type series of the form $(*)$ with $a(k)$ not a product of three nontrivial parts were found (cf. [3,4,9,29]). For example, H. H. Chan, S. H. Chan and Z. Liu [3] proved that

where $D_n$ denotes the Domb number $\sum_{k = 0}^n \binom nk^2 \binom{2k}k \binom{2(n-k)}{n-k}$; Zudilin [53] conjectured its $p$-adic analogue:

The author [45,Conjecture 77] conjectured further that

for each odd prime $p$ and positive integer $n$.

Let $b,c\in{\Bbb Z}$. For each $n\in{\Bbb N} = \{0,1,2,\ldots\}$, we denote the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$ by $T_n(b,c)$, and call it a generalized central trinomial coefficient. In view of the multinomial theorm, we have

Note also that

and

for all $n\in{\Bbb Z}^+.$ Clearly, $T_n(2,1) = \binom{2n}n$ for all $n\in{\Bbb N}$. Those $T_n: = T_n(1,1)$ with $n\in{\Bbb N}$ are the usual central trinomial coefficients, and they play important roles in enumerative combinatorics. We view $T_n(b,c)$ as a natural generalization of central binomial and central trinomial coefficients.

For $n\in{\Bbb N}$ the Legendre polynomial of degree $n$ is defined by

It is well-known that if $b,c\in{\Bbb Z}$ and $b^2-4c\not = 0$ then

Via the Laplace-Heine asymptotic formula for Legendre polynomials, for any positive real numbers $b$ and $c$ we have

(cf. [40]). For any real numbers $b$ and $c<0$, S. Wagner [48] confirmed the author's conjecture that

In 2011, the author posed over 60 conjectural series for $1/\pi$ of the following new types with $a,b,c,d,m$ integers and $mbcd(b^2-4c)$ nonzero (cf. Sun [34,40]).

Type Ⅰ. $\sum_{k = 0}^\infty \frac{a+dk}{m^k} \binom{2k}k^2T_k(b,c)$.

Type Ⅱ. $\sum_{k = 0}^\infty \frac{a+dk}{m^k} \binom{2k}k \binom{3k}kT_k(b,c)$.

Type Ⅲ. $\sum_{k = 0}^\infty \frac{a+dk}{m^k} \binom{4k}{2k} \binom{2k}kT_k(b,c)$.

Type Ⅳ. $\sum_{k = 0}^\infty \frac{a+dk}{m^k} \binom{2k}{k}^2T_{2k}(b,c)$.

Type Ⅴ. $\sum_{k = 0}^\infty \frac{a+dk}{m^k} \binom{2k}{k} \binom{3k}kT_{3k}(b,c)$.

Type Ⅵ. $\sum_{k = 0}^\infty \frac{a+dk}{m^k}T_{k}(b,c)^3,$

Type Ⅶ. $\sum_{k = 0}^\infty \frac{a+dk}{m^k} \binom{2k}kT_{k}(b,c)^2,$

In general, the corresponding $p$-adic congruences of these seven-type series involve linear combinations of two Legendre symbols. The author's conjectural series of types Ⅰ-Ⅴ and Ⅶ were studied in [6,49,54]. The author's three conjectural series of type Ⅵ and two series of type Ⅶ remain open. For example, the author conjectured that

as well as its $p$-adic analogue

where $p$ is any prime greater than $3$.

In 1905, J. W. L. Glaisher [15] proved that

This actually follows from the following finite identity observed by the author [38]:

Motivated by Glaisher's identity and Ramanujan-type series for $1/\pi$, we obtain the following theorem.

Theorem 1.1. We have the following identities:

and

where

Recall that the Catalan numbers are given by

For $k\in{\Bbb N}$ it is easy to see that

Thus, for any $a,b,c,m\in{\Bbb Z}$ with $|m|{\geq}64$, we have

For example, (1.2) has the equivalent form

For any odd prime $p$, the congruence (1.4) of V.J.W. Guo and J.-C. Liu [19] has the equivalent form

(where $E_0,E_1,\ldots$ are the Euler numbers), and we note that this is also equivalent to the congruence

Recently, C. Wang [50] proved that for any prime $p>3$ we have

and

(Actually, Wang stated his results only in the language of hypergeometric series.) These two congruences extend a conjecture of Guo and M. J. Schlosser [21].

We are also able to prove some other variants of Ramanujan-type series such as

and

Now we state our second theorem.

Theorem 1.2. We have the identities

where

For $k = j+1\in{\Bbb Z}^+$, it is easy to see that

Thus, for any $a,b,c,m\in{\Bbb Z}$ with $0<|m| \leq 64$, we have

For example, (1.77) has the following equivalent form

In contrast with the Domb numbers, we introduce a new kind of numbers

The values of $S_n\ (n = 0,\ldots,10)$ are

respectively. We may extend the numbers $S_n\ (n\in{\Bbb N})$ further. For $b,c\in{\Bbb Z}$, we define

Note that $S_n(1,1) = S_n$ and $S_n(2,1) = D_n$ for all $n\in{\Bbb N}$.

Now we state our third theorem.

Theorem 1.3. We have

Remark 1.1. The author found the 10 series in Theorem 1.3 in Nov. 2019.

We shall prove Theorems 1.1-1.3 in the next section. In Sections 3-10, we propose 117 new conjectural series for powers of $\pi$ involving generalized central trinomial coefficients. In particular, we will present in Section 3 four conjectural series for $1/\pi$ of the following new type:

Type Ⅷ. $\sum_{k = 0}^\infty \frac{a+dk}{m^k}T_k(b,c)T_{k}(b_*,c_*)^2,$

where $a,b,b_*,c,c_*,d,m$ are integers with $mbb_*cc_*d(b^2-4c)(b_*^2-4c_*)(b^2c_*-b_*^2c)\not = 0$.

Unlike Ramanujan-type series given by others, all our series for $1/\pi$ of types Ⅰ-Ⅷ have the general term involving a product of three generalized central trinomial coefficients.

Motivated by the author's effective way to find new series for $1/\pi$ (cf. Sun [35]), we formulate the following general characterization of rational Ramanujan-type series for $1/\pi$ via congruences.

Conjecture 1.1 (General Criterion for Rational Ramanujan-type Series for $1/\pi$). Suppose that the series $\sum_{k = 0}^\infty \frac{bk+c}{m^k}a_k$ converges, where $(a_k)_{k{\geq}0}$ is an integer sequence and $b,c,m$ are integers with $bcm\not = 0$. Suppose also that there are no $a,x \in \mathbb{Z}$ such that ${a_n} = a(_n^{2n})\sum\nolimits_{k = 0}^n ( _k^{2k}{)^2}(_{n - k}^{2(n - k)}){x^{n - k}}$ for all $n \in \mathbb{N}$. Let $r\in\{1,2,3\}$ and let ${d_1}, \ldots ,{d_r} \in {\mathbb{Z}^ + }$ with $\sqrt{d_i/d_j}$ irrational for all distinct $i,j\in\{1,\ldots,r\}$. Then

for some nonzero rational numbers $\lambda_1,\ldots,\lambda_r$ if and only if there are positive integers $d_j\ (r<j \leq 3)$ and rational numbers $c_1,c_2,c_3$ with $\prod_{i = 1}^rc_i\not = 0$, such that for any prime $p>3$ with $p\nmid m\prod_{i = 1}^rd_i$ and $c_1,c_2,c_3\in{\Bbb Z}_p$ we have

where ${\varepsilon}_i\in\{\pm1\}$, ${\varepsilon}_i = -1$ if $d_i$ is not an integer square, and $c_2 = c_3 = 0$ if $r = 1$ and ${\varepsilon}_1 = 1$.

For a Ramanujan-type series of the form (1.98), we call $r$ its rank. We believe that there are some Ramanujan-type series of rank three but we have not yet found such a series.

Conjecture 1.2. Let $(a_n)_{n{\geq}0}$ be an integer sequence with no $a,x \in \mathbb{Z}$ such that ${a_n} = a(_n^{2n})\sum\nolimits_{k = 0}^n ( _k^{2k}{)^2}(_{n - k}^{2(n - k)}){x^{n - k}}$ for all $n \in \mathbb{N}$, and let $b,c,m,d_1,d_2,d_3\in{\Bbb Z}$ with $bcm\not = 0$. Assume that $\lim_{n\to+\infty}\root n\of{|a_n|} = r<|m|$, and $\pi\sum_{k = 0}^\infty \frac{bk+c}{m^k}a_k$ is an algebraic number. Suppose that $c_1,c_2,c_3 \in \mathbb{Q}$ with $c_1+c_2+c_3 = a_0c$ , and

for all primes $p>3$ with $p\nmid d_1d_2d_3m$ and $c_1,c_2,c_3\in{\Bbb Z}_p$. Then, for any prime $p>3$ with $p\nmid m$, $c_1,c_2,c_3\in{\Bbb Z}_p$ and $( \frac{d_1}p) = ( \frac{d_2}p) = ( \frac{d_3}p) = {\delta}\in\{\pm1\}$, we have

Joint with the author's PhD student Chen Wang, we pose the following conjecture.

Conjecture 1.3 (Chen Wang and Z.-W. Sun). Let $(a_k)_{k{\geq}0}$ be an integer sequence with $a_0 = 1$. Let $b,c,m,d_1,d_2,d_3\in{\Bbb Z}$ with $bm\not = 0$, and let $c_1,c_2,c_3$ be rational numbers. If $\pi\sum_{k = 0}^\infty \frac{bk+c}{m^k}a_k$ is an algebraic number, and the congruence (1.100) holds for all primes $p>3$ with $p\nmid d_1d_2d_3m$ and $c_1,c_2,c_3\in{\Bbb Z}_p$, then we must have $c_1+c_2+c_3 = c.$

Remark 1.2. The author [39,Conjecture 1.1(i)] conjectured that

for any prime $p>3$, which was confirmed by Y.-P. Mu and Z.-W. Sun [26]. This is not a counterexample to Conjecture 1.3 since $\sum_{k = 0}^\infty(8k+5)T_k^2$ diverges.

All the new series and related congruences in Sections 3-9 support Conjectures 1.1-1.3. We discover the conjectural series for $1/\pi$ in Sections 3-9 based on the author's previous $\mathsf{Philosophy about Series for}$ $1/\pi$} stated in [35], the PSLQ algorithm to discover integer relations (cf. [13]), and the following $\mathsf{Duality Principle}$ based on the author's experience and intuition.

Conjecture 1.4 (Duality Principle). Let $(a_k)_{k{\geq}0}$ be an integer sequence such that

for any prime $p\nmid 6dD$ and $k\in\{0,\ldots,p-1\}$, where $d$ and $D$ are fixed nonzero integers. If $a_0,a_1,\ldots$ are not all zero and $m$ is a nonzero integer such that

for some $b,d_1,d_2,d_3\in{\Bbb Z}^+$, $c\in{\Bbb Z}$ and $\lambda_1,\lambda_2,\lambda_3\in{\Bbb Q}$, then $m$ divides $D$, and

for any prime $p>3$ with $p\nmid dD$.

Remark 1.3 (ⅰ) For any prime $p>3$ with $p\nmid dDm$, the congruence (1.102) holds modulo $p$ by (1.101) and Fermat's little theorem. We call $\sum_{k = 0}^{p-1}a_k/( \frac Dm)^k$ the dual of the sum $\sum_{k = 0}^{p-1}a_k/m^k$.

(ⅱ) For any $b,c\in{\Bbb Z}$ and odd prime $p\nmid b^2-4c$, it is known (see, e.g., [39,Lemma 2.2]) that

for all $k = 0,1,\ldots,p-1$.

For a series $\sum_{k = 0}^\infty a_k$ with $a_0,a_1,\ldots$ real numbers, if $\lim_{k\to+\infty}a_{k+1}/a_k = r\in(-1,1)$ then we say that the series converge at a geometric rate with ratio $r$. Except for $(7.1)$, all other conjectural series in Sections 3-9 converge at geometric rates and thus one can easily check them numerically via a computer.

In Section 10, we pose two curious conjectural series for $\pi$ involving the central trinomial coefficients.

2.   Proofs of Theorems 1.1-1.3

Lemma 2.1. Let $m\not = 0$ and $n{\geq}0$ be integers. Then

Remark 2.1. The eight identities in Lemma 2.1 can be easily proved by induction on $n$. In light of Stirling's formula, $n!\sim\sqrt{2\pi n}(n/e)^n$ as $n\to+\infty$, we have

Proof of Theorem 1.1. Just apply Lemma 2.1 and the 36 known rational Ramanujan-type series listed in [16]. Let us illustrate the proofs by showing (1.1), (1.2), (1.71) and (1.72) in details.

By (2.1) with $m = -64$, we have

Note that

and recall Bauer's series

So, we get

This proves (1.1). By (2.2) with $m = -64$, we have

and hence

Combining this with $(1.1)$ we immediately get $(1.2)$.

In view of (2.7) with $m = -640320^3$, we have

and hence

In 1987, D. V. Chudnovsky and G. V. Chudnovsky [8] got the formula

which enabled them to hold the world record for the calculation of $\pi$ during 1989–1994. Note that

and hence

This proves $(1.71)$.

By (2.8) with $m = -640320^3$, we have

and hence

Note that

Therefore, with the help of $(1.71)$ we get

This proves $(1.72)$.

The identities (1.3)–(1.70) can be proved similarly.

Lemma 2.2. Let $m$ and $n>0$ be integers. Then

Remark 2.2. This can be easily proved by induction on $n$.

Proof of Theorem 1.2. We just apply Lemma 2.2 and use the known identities:

Here, the first identity was found and proved by D. Zeilberger [52] in 1993. The second, third and fourth identities were obtained by J. Guillera [17] in 2008. The fifth identity on $K$ was conjectured by Sun [33] and later confirmed by K. Hessami Pilehrood and T. Hessami Pilehrood [22] in 2012. The last four identities were also conjectured by Sun [33], and they were later proved in the paper [18,Theorem 3] by Guillera and M. Rogers.

Let us illustrate our proofs by proving (1.77)-(1.79) and (1.82)-(1.83) in details.

In view of (2.11) with $m = 16$, we have

for all $n\in{\Bbb Z}^+$, and hence

Notice that

So we have

and hence (1.77) holds.

By (2.11) with $m = -64$, we have

for all $n\in{\Bbb Z}^+$, and hence

Since $16k^3+4k^2-2k-1 = (4k-1)(2k+1)^2-2k(4k+1)$ and

we see that

and hence (1.78) holds. In light of (2.12) with $m = -64$, we have

for all $n\in{\Bbb Z}^+$, and hence

Since $16k^3+12k^2+6k+1 = 2k(2k+1)(4k+1)+(4k+1)$, with the aid of (1.78) we obtain

This proves (1.79).

By (2.13) with $m = -27$, we have

As

and

we see that (1.82) follows. By (2.14) with $m = -27$, we have

and hence

As

with the aid of (1.82) we get

and hence $(1.83)$ follows.

Other identities in Theorem 1.2 can be proved similarly.

For integers $n{\geq} k{\geq}0$, we define

For $n\in{\Bbb N}$ we set

Lemma 2.3. For any $n\in{\Bbb N}$, we have

and

where $f_n$ denotes the Franel number $\sum_{k = 0}^n \binom nk^3$.

Proof. For $n,i,k\in{\Bbb N}$ with $i \leq k$, we set

By the telescoping method for double summation [7], for

with $0 \leq i \leq k$, we find that

where

and

with $(-1)!,(-2)!,\ldots$ regarded as $+\infty$, and $p(n,i,k)$ and $q(n,i,k)$ given by

and

respectively. Therefore

and hence

satisfies the recurrence relation

As pointed out by J. Franel [14], the Franel numbers satisfy the same recurrence. Note also that $u(0) = f_0 = 1$ and $u(1) = f_1 = 2$. So we always have $u(n) = f_n$. This proves (2.19).

The identity (2.20) can be proved similarly. In fact, if we use $v(n)$ denote the left-hand side or the right-hand side of (2.20), then we have the recurrence

In view of the above, we have completed the proof of Lemma 2.3.

Lemma 2.4. For any $c\in{\Bbb Z}$ and $n\in{\Bbb N}$, we have

Proof. For each $k = 0,\ldots,n$, we have

If $i,j\in{\Bbb N}$ and $i+j = r \leq n/2$, then

with the aid of the Chu-Vandermonde identity. Therefore

This proves (2.21).

Lemma 2.5. For $k\in{\Bbb N}$ and $l\in{\Bbb Z}^+$, we have

Proof. Let $n = k+l$. Then

and

Hence

This proves (2.22).

To prove Theorem 1.3, we need an auxiliary theorem.

Theorem 2.6. Let $a$ and $b$ be real numbers. For any integer $m$ with $|m|{\geq}94$, we have

Proof. Let $N\in{\Bbb N}$. In view of (2.21),

and similarly

where we consider $\binom x{-1}$ as $0$.

If $l$ is an integer in the interval $(N/2,N]$, then by Lemma 2.5 we have

where $P_l(x)$ is the Legendre polynomial of degree $l$. Thus

and

Recall that

As $|m|{\geq}94$, we have $|m|>16(3+2\sqrt2)\approx 93.255$ and hence

converges. Thus

and hence by the above we have

and

Therefore, with the aid of (2.19), we obtain

and

In view of (2.25) and (2.20),

Combining this with (2.24), we immediately obtain the desired (2.23).

Proof of Theorem 1.3. Let $a,b,m\in{\Bbb Z}$ with $|m|{\geq}6$. Since

for any $n\in{\Bbb N}$, we have $4^nS_n(1,m) = S_n(4,16m)$ for all $n\in{\Bbb N}$. Thus, in light of Theorem 2.6,

Therefore

It is known (cf. [5,4]) that

So we get the identities (1.88)-(1.97) finally.

3.   New series involving $T_n(b,c)$ for $1/\pi$ related to types Ⅰ-Ⅷ

Now we pose a conjecture related to the series (Ⅰ1)-(Ⅰ4) of Sun [34,40].

Conjecture 3.1. We have the following identities:

Remark 3.1. For each $k\in{\Bbb N}$, we have

since $\binom{2k}k = (k+1)C_k$ and $\binom{2k}{k+1} = kC_k$. Thus, for example, [40,(I1)] and (I1$'$) together imply that

and (I5) and (I5$'$) imply that

For the conjectural identities in Conjecture 3.1, we have conjectures for the corresponding $p$-adic congruences. For example, in contrast with (I2$'$), we conjecture that for any prime $p>3$ we have the congruences

and

Concerning (I5) and (I5$''$), we conjecture that

and

for each $n = 2,3,\ldots$, and that for any prime $p{\equiv}1\ ({\rm{mod}}\ 4)$ with $p = x^2+4y^2\ (x,y\in{\Bbb Z})$ we have

and

By [40,Theorem 5.1], we have

for any prime $p{\equiv}3\ ({\rm{mod}}\ 4)$. The identities (I5), (I5$'$) and (I5$''$) were formulated by the author on Dec. 9, 2019.

Next we pose a conjecture related to the series (Ⅱ1)-(Ⅱ7) and (Ⅱ10)-(Ⅱ12) of Sun [34,40].

Conjecture 3.2. We have the following identities:

Remark 3.2. We also have conjectures on related congruences. For example, concerning (Ⅱ), for any prime $p>3$ we conjecture that

and that

where $x$ and $y$ are integers. The identities (Ⅱ13), (Ⅱ13$'$), (Ⅱ14) and (Ⅱ14$'$) were found by the author on Dec. 11, 2019.

The following conjecture is related to the series (Ⅲ1)-(Ⅲ10) and (Ⅲ12) of Sun [34,40].

Conjecture 3.3. We have the following identities:

and

The following conjecture is related to the series (Ⅳ1)-(Ⅳ21) of Sun [34,40].

Conjecture 3.4. We have the following identities:

For the five open conjectural series (Ⅵ1), (Ⅵ2), (Ⅵ3), (ⅥI2) and (ⅥI7) of Sun [34,40], we make the following conjecture on related supercongruences.

Conjecture 3.5. Let $p$ be an odd prime and let $n\in{\Bbb Z}^+$. If $( \frac 3p) = 1$, then

divided by $(pn)^2$ is a $p$-adic integer. If $p\not = 5$, then

divided by $(pn)^2$ is a $p$-adic integer. If $( \frac 7p) = 1$ but $p\not = 3$, then

divided by $(pn)^2$ is a $p$-adic integer. If $p{\equiv}\pm1\ ({\rm{mod}}\ 8)$ but $p\not = 7$, then

divided by $(pn)^2$ is a $p$-adic integer. If $( \frac{-6}p) = 1$ but $p\not = 7,31$, then

divided by $(pn)^2$ is a $p$-adic integer.

Now we pose four conjectural series for $1/\pi$ of type Ⅷ.

Conjecture 3.6. We have

Remark 3.3. The author found the identity (Ⅷ1) on Nov. 3, 2019. The identities (Ⅷ2), (Ⅷ3) and (Ⅷ4) were formulated on Nov. 4, 2019.

Below we present some conjectures on congruences related to Conjecture 3.6.

Conjecture 3.7. (ⅰ) For each $n\in{\Bbb Z}^+$, we have

and this number is odd if and only if $n$ is a power of two $($i.e., $n\in\{2^a:\ a\in{\Bbb N}\})$.

(ⅱ) Let $p\not = 2,5$ be a prime. Then

If $( \frac 3p) = ( \frac{-5}p) = 1$, then

for all $n\in{\Bbb Z}^+$.

(ⅲ) Let $p\not = 2,5$ be a prime. Then

Remark 3.4. The imaginary quadratic field ${\Bbb Q}(\sqrt{-5})$ has class number two.

Conjecture 3.8. (ⅰ) For any $n\in{\Bbb Z}^+$, we have

and the number is odd if and only if $n$ is a power of two.

(ⅱ) Let $p>3$ be a prime. Then

If $( \frac 3p) = ( \frac{-5}p) = 1$, then

for all $n\in{\Bbb Z}^+$.

(ⅲ) Let $p>5$ be a prime. Then

Remark 3.5. This conjecture can be viewed as the dual of Conjecture 3.7. Note that the series $\sum_{k = 0}^\infty \frac{(40k+27}{(-6)^k}T_k(4,1)T_k(1,-1)^2$ diverges.

Conjecture 3.9. (ⅰ) For each $n\in{\Bbb Z}^+$, we have

(ⅱ) Let $p>3$ be a prime. Then

If $p{\equiv}1,9\ ({\rm{mod}}\ {20})$, then

divided by $(pn)^2$ is a $p$-adic integer for each $n\in{\Bbb Z}^+$.

(ⅲ) Let $p>5$ be a prime. Then

Remark 3.6. The imaginary quadratic field ${\Bbb Q}(\sqrt{-15})$ has class number two.

Conjecture 3.10. (ⅰ) For each $n\in{\Bbb Z}^+$, we have

(ⅱ) Let $p>5$ be a prime. Then

If $p{\equiv}1,9\ ({\rm{mod}}\ {20})$, then

divided by $(pn)^2$ is a $p$-adic integer for each $n\in{\Bbb Z}^+$.

(ⅲ) Let $p>5$ be a prime. Then

Remark 3.7. This conjecture can be viewed as the dual of Conjecture 3.9. Note that the series

diverges.

Conjecture 3.11. (ⅰ) For each $n\in{\Bbb Z}^+$, we have

(ⅱ) Let $p>3$ be a prime. Then

If $( \frac {-1}p) = ( \frac {15}p) = 1$, then

divided by $(pn)^2$ is an $p$-adic integer for any $n\in{\Bbb Z}^+$.

(ⅲ Let $p>7$ be a prime. Then

where $x$ and $y$ are integers.

Remark 3.8. Note that the imaginary quadratic field ${\Bbb Q}(\sqrt{-105})$ has class number $8$.

Conjecture 3.12. (ⅰ) For each $n\in{\Bbb Z}^+$, we have

and this number is odd if and only if $n$ is a power of two.

(ⅱ) Let $p>5$ be a prime. Then

If $( \frac 3p) = ( \frac{-5}p) = 1$, then

divided by $(pn)^2$ is a $p$-adic integer for each $n\in{\Bbb Z}^+$.

(ⅲ) Let $p>5$ be a prime with $p\not = 11$. Then

where $x$ and $y$ are integers.

Remark 3.9. Note that the imaginary quadratic field ${\Bbb Q}(\sqrt{-165})$ has class number $8$.

4.   Congruences related to Theorem 1.3

Conjectures 4.1–4.14 below provide congruences related to (1.88)–(1.97).

Conjecture 4.1. (ⅰ) For any $n\in{\Bbb Z}^+$, we have

(ⅱ) Let $p>3$ be a prime. Then

If $p{\equiv}1\ ({\rm{mod}}\ 3)$, then

for all $n\in{\Bbb Z}^+$.

(ⅲ) For any prime $p>3$, we have

Conjecture 4.2. (ⅰ) For any $n\in{\Bbb Z}^+$, we have

and this number is odd if and only if $n$ is a power of two.

(ⅱ) Let $p\not = 7$ be an odd prime. Then

and moreover

for all $n\in{\Bbb Z}^+$.

(ⅲ) For any prime $p\not = 2,7$, we have

where $x$ and $y$ are integers.

Conjecture 4.3. (ⅰ) For any $n\in{\Bbb Z}^+$, we have

and this number is odd if and only if $n$ is a power of two.

(ⅱ) Let $p$ be an odd prime with $p\not = 5$. Then

If $p{\equiv}1\ ({\rm{mod}}\ 3)$, then

for all $n\in{\Bbb Z}^+$.

(ⅲ) For any prime $p\not = 2,5$, we have

where $x$ and $y$ are integers.

Conjecture 4.4. (ⅰ) For any $n\in{\Bbb Z}^+$, we have

(ⅱ) Let $p\not = 5$ be an odd prime. Then

for all $n\in{\Bbb Z}^+$.

(ⅲ) For any prime $p>3$ with $p\not = 11$, we have

where $x$ and $y$ are integers.

Conjecture 4.5. (ⅰ) For any $n\in{\Bbb Z}^+$, we have

and this number is odd if and only if $n$ is a power of two.

(ⅱ) Let $p$ be an odd prime with $p\not = 7$. Then

If $p{\equiv}1,3\ ({\rm{mod}}\ 8)$, then

for all $n\in{\Bbb Z}^+$.

(ⅲ) For any prime $p\not = 2,7$, we have

where $x$ and $y$ are integers.

Conjecture 4.6. (ⅰ) For any $n\in{\Bbb Z}^+$, we have

(ⅱ) Let $p\not = 13$ be an odd prime. Then

divided by $(pn)^2$ is a $p$-adic integer for any $n\in{\Bbb Z}^+$.

(ⅲ) For any prime $p>3$ with $p\not = 19$, we have

where $x$ and $y$ are integers.

Conjecture 4.7. (ⅰ) For any $n\in{\Bbb Z}^+$, we have

and this number is odd if and only if $n\in\{2^a:\ a\in{\Bbb N}\}$.

(ⅱ) Let $p$ be an odd prime with $p\not = 5,13$. Then

If $( \frac{-6}p) = 1$, then

divided by $(pn)^2$ is a $p$-adic integer for any $n\in{\Bbb Z}^+$.

(ⅲ)For any odd prime $p\not = 5,13$, we have

where $x$ and $y$ are integers.

Conjecture 4.8. (ⅰ) For any $n\in{\Bbb Z}^+$, we have

and this number is odd if and only if $n\in\{2^a:\ a\in{\Bbb N}\}$.

(ⅱ) Let $p\not = 7,31$ be an odd prime. Then

divided by $(pn)^2$ is a $p$-adic integer for any $n\in{\Bbb Z}^+$.

(ⅲ) For any prime $p>3$ with $p\not = 7,31$, we have

where $x$ and $y$ are integers.

Conjecture 4.9. (ⅰ) For any $n\in{\Bbb Z}^+$, we have

(ⅱ) Let $p\not = 5,7$ be an odd prime. Then

If $p{\equiv}1\ ({\rm{mod}}\ 3)$, then

divided by $(pn)^2$ is a $p$-adic integer for any $n\in{\Bbb Z}^+$.

(ⅲ) For any prime $p>7$ with $p\not = 17$, we have

where $x$ and $y$ are integers.

Conjecture 4.10. (ⅰ) For any $n\in{\Bbb Z}^+$, we have

$\rm(ⅱ)$ Let $p\not = 5,53$ be an odd prime. Then

divided by $(pn)^2$ is a $p$-adic integer for any $n\in{\Bbb Z}^+$.

$\rm(ⅲ)$ For any prime $p>5$ with $p\not = 59$, we have

where $x$ and $y$ are integers.

Conjecture 4.11. For any odd prime $p$,

Also, for any prime $p{\equiv}1\ ({\rm{mod}}\ 4)$ we have

Conjecture 4.12. $\rm(ⅰ)$ For any $n\in{\Bbb Z}^+$, we have

and this number is odd if and only if $n$ is a power of two.

$\rm(ⅱ)$ For any odd prime $p$ and positive integer $n$, we have

$\rm(ⅲ)$ Let $p$ be an odd prime. Then

Conjecture 4.13. $\rm(ⅰ)$ For any $n\in{\Bbb Z}^+$, we have

and this number is odd if and only if $n$ is a power of two.

$\rm(ⅱ)$ Let $p>3$ be a prime. Then

If $p{\equiv}\pm1\ ({\rm{mod}}\ {12})$, then

for all $n\in{\Bbb Z}^+$.

$\rm(ⅲ)$ For any prime $p>3$, we have

Conjecture 4.14. $\rm(ⅰ)$ For any $n\in{\Bbb Z}^+$, we have

$\rm(ⅱ)$ Let $p$ be an odd prime. Then

If $p{\equiv}1\ ({\rm{mod}}\ 3)$, then

for all $n\in{\Bbb Z}^+$.

$\rm(ⅲ)$ For any prime $p>3$, we have

where $x$ and $y$ are integers.

Conjecture 4.15. Let $p$ be an odd prime with $p\not = 5$. Then

where $x$ and $y$ are integers. If $( \frac{-5}p) = 1$, then

Remark 4.1. We also have some similar conjectures involving

modulo $p^2$, where $p$ is a prime greater than $3$.

Motivated by Theorem 2.6, we pose the following general conjecture.

Conjecture 4.16. For any odd prime $p$ and integer $m\not{\equiv}0\ ({\rm{mod}}\ p)$, we have

and

Remark 4.2 We have checked this conjecture via $\mathsf{Mathematica}$. In view of the proof of Theorem 2.6, both (4.51) and (4.52) hold modulo $p$.

5.   Series for $1/\pi$ involving $T_n(b,c)$ and $Z_n=\sum_{k=0}^n \binom nk \binom{2k}k \binom{2(n-k)}{n-k}$

The numbers

were first introduced by D. Zagier in his paper [51] the preprint of which was released in 2002. Thus we name such numbers as Zagier numbers. As pointed out by the author [41,Remark 4.3], for any $n\in{\Bbb N}$ the number $2^nZ_n$ coincides with the so-called CLF (Catalan-Larcombe-French) number

Let $p$ be an odd prime. For any $k = 0,\ldots,p-1$, we have

by F. Jarvis and H.A. Verrill [24,Corollary 2.2], and hence

Combining this with Remark 1.3(ⅱ), we see that

for any $b,c,m\in{\Bbb Z}$ with $p\nmid (b^2-4c)m$.

J. Wan and Zudilin [49] obtained the following irrational series for $1/\pi$ involving the Legendre polynomials and the Zagier numbers:

Via our congruence approach (including Conjecture 1.4), we find 24 rational series for $1/\pi$ involving $T_n(b,c)$ and the Zagier numbers. Theorem 1 of [49] might be helpful to solve some of them.

Conjecture 5.1. We have the following identities for $1/\pi$.

Below we present some conjectures on congruences related to $(5.1)$, $(5.2)$, $(5.4)$ and $(5.9)$.

Conjecture 5.2. (ⅰ) For any $n\in{\Bbb Z}^+$, we have

$\rm (ⅱ)$ Let $p$ be an odd prime with $p\not = 5$. Then

If $p{\equiv}\pm1\ ({\rm{mod}}\ {5})$, then

for all $n\in{\Bbb Z}^+$.

$\rm (ⅲ)$ For any prime $p>5$, we have

Conjecture 5.3. (ⅰ) For any $n\in{\Bbb Z}^+$, we have

$\rm (ⅱ)$ Let $p>3$ be a prime with $p\not = 7$. Then

If $( \frac 7p) = 1$, then

for all $n\in{\Bbb Z}^+$.

$\rm (ⅲ)$ For any prime $p>3$ with $p\not = 7$, we have

Conjecture 5.4. $\rm(i)$ For any $n\in{\Bbb Z}^+$, we have

$\rm (ⅱ)$ Let $p>3$ be a prime. Then

If $p{\equiv}1,3\ ({\rm{mod}}\ 8)$, then

for all $n\in{\Bbb Z}^+$.

$\rm (ⅲ)$ For any prime $p>3$, we have

Conjecture 5.5. $\rm(ⅰ)$ For any $n\in{\Bbb Z}^+$, we have

$\rm (ⅱ)$ Let $p$ be an odd prime with $p\not = 5$. Then

If $( \frac{-15}p) = 1$, then

for all $n\in{\Bbb Z}^+$.

$\rm (ⅲ)$ For any prime $p>5$, we have

where $x$ and $y$ are integers.

6.   Series for $1/\pi$ involving $T_k(b,c)$ and the Franel numbers

Sun [36,37] obtained some supercongruences involving the Franel numbers $f_n = \sum_{k = 0}^n \binom nk^3\ (n\in{\Bbb N})$. M. Rogers and A. Straub [30] confirmed the $520$-series for $1/\pi$ involving Franel polynomials conjectured by Sun [34].

Let $p$ be an odd prime. By [24,Lemma 2.6], we have $f_k{\equiv}(-8)^kf_{p-1-k}\ ({\rm{mod}}\ p)$ for each $k = 0,\ldots,p-1$. Combining this with Remark 1.3(ⅱ), we see that

for any $b,c,m\in{\Bbb Z}$ with $p\nmid (b^2-4c)m$.

Wan and Zudilin [49] deduced the following irrational series for $1/\pi$ involving the Legendre polynomials and the Franel numbers:

Via our congruence approach (including Conjecture 1.4), we find $12$ rational series for $1/\pi$ involving $T_n(b,c)$ and the Franel numbers; Theorem 1 of [49] might be helpful to solve some of them.

Conjecture 6.1. We have

We now present a conjecture on congruence related to $(6.3)$.

Conjecture 6.2. $\rm(ⅰ)$ For any $n\in{\Bbb Z}^+$, we have

$\rm (ⅱ)$ Let $p>5$ be a prime. Then

If $( \frac{-5}p) = 1$, then

for all $n\in{\Bbb Z}^+$.

$\rm (ⅲ)$ For any prime $p>5$, we have

Remark 6.1 This conjecture was formulated by the author on Oct. 25, 2019.

Conjecture 6.3. For any $n\in{\Bbb Z}^+$, we have

$\rm (ⅱ)$ Let $p$ be an odd prime. Then

If $( \frac{-5}p) = 1$, then

divided by $(pn)^2$ is a $p$-adic integer for any $n\in{\Bbb Z}^+$.

$\rm (ⅲ)$ Let $p>5$ be a prime. Then

Remark 6.2. This conjecture is the dual of Conjecture 6.2.

The following conjecture is related to the identity $(6.8)$.

Conjecture 6.4. $\rm(ⅰ)$ For any $n\in{\Bbb Z}^+$, we have

$\rm (ⅱ)$ Let $p>2$ be a prime with $p\not = 7,11$. Then

If $( \frac {-39}p) = 1$, then

divided by $(pn)^2$ is a $p$-adic integer for any $n\in{\Bbb Z}^+$.

$\rm (ⅲ)$ Let $p>3$ be a prime with $p\not = 7,11,13$. Then

where $x$ and $y$ are integers.

Remark 6.3. Note that the imaginary quadratic field ${\Bbb Q}(\sqrt{-273})$ has class number $8$.

The following conjecture is related to the identity $(6.10)$.

Conjecture 6.5. $\rm(ⅰ)$ For any $n\in{\Bbb Z}^+$, we have

$\rm (ⅱ)$ Let $p>3$ be a prime with $p\not = 23$. Then

If $( \frac p{23}) = 1$, then

divided by $(pn)^2$ is a $p$-adic integer for any $n\in{\Bbb Z}^+$.

$\rm (ⅲ)$ Let $p>3$ be a prime with $p\not = 23$. Then

where $x$ and $y$ are integers.

Remark 6.4. Note that the imaginary quadratic field ${\Bbb Q}(\sqrt{-345})$ has class number $8$.

The following conjecture is related to the identity $(6.12)$.

Conjecture 6.6. $\rm(ⅰ)$ For any $n\in{\Bbb Z}^+$, we have

$\rm (ⅱ)$ Let $p$ be an odd prime with $p\not = 11,13$. Then

If $( \frac {-14}p) = 1$, then

divided by $(pn)^2$ is a $p$-adic integer for any $n\in{\Bbb Z}^+$.

$\rm (ⅲ)$ Let $p>3$ be a prime with $p\not = 7,11,13$. Then

where $x$ and $y$ are integers.

Remark 6.5. Note that the imaginary quadratic field ${\Bbb Q}(\sqrt{-462})$ has class number $8$.

The identities $(6.5),\, (6.6),\,(6.7),\,(6.9),\,(6.11)$ are related to the quadratic fields

(with class number $8$) respectively. We also have conjectures on related congruences similar to Conjectures 6.4, 6.5 and 6.6.

7.   Series for $1/\pi$ involving $T_n(b,c)$ and $g_n=\sum_{k=0}^n \binom nk^2 \binom{2k}k$

For $n\in{\Bbb N}$ let

It is known that $g_n = \sum_{k = 0}^n \binom nk f_k$ for all $n\in{\Bbb N}$. See [43,20,26] for some congruences on polynomials related to these numbers.

Let $p>3$ be a prime. For any $k = 0,\ldots,p-1$, we have

by [24,Lemma 2.7(ⅱ)]. Combining this with Remark 1.3(ⅱ), we see that

for any $b,c,m\in{\Bbb Z}$ with $p\nmid (b^2-4c)m$.

Wan and Zudilin [49] obtained the following irrational series for $1/\pi$ involving the Legendre polynomials and the sequence $(g_n)_{n{\geq}0}$:

Using our congruence approach (including Conjecture 1.4), we find 12 rational series for $1/\pi$ involving $T_n(b,c)$ and $g_n$; Theorem 1 of [49] might be helpful to solve some of them.

Conjecture 7.1. We have the following identities.

Now we present a conjecture on congruences related to $(7.6)$.

Conjecture 7.2. $\rm(ⅰ)$ For any $n\in{\Bbb Z}^+$, we have

and this number is odd if and only if $n\in\{2^a:\ a\in{\Bbb N}\}$.

$\rm(ⅱ)$ Let $p>7$ be a prime. Then

If $p{\equiv}1,3\ ({\rm{mod}}\ 8)$, then

divided by $(pn)^2$ is a $p$-adic integer for any $n\in{\Bbb Z}^+$.

$\rm(ⅲ)$ Let $p>7$ be a prime. Then

where $x$ and $y$ are integers.

Remark 7.1. Note that the imaginary quadratic field ${\Bbb Q}(\sqrt{-210})$ has class number $8$.

The following conjecture is related to the identity $(7.8)$.

Conjecture 7.3. $\rm(ⅰ)$ For any $n\in{\Bbb Z}^+$, we have

and this number is odd if and only if $n\in\{2^a:\ a\in{\Bbb N}\}$.

$\rm(ⅱ)$ Let $p>7$ be a prime. Then

If $( \frac p3) = ( \frac p{5})$, then

divided by $(pn)^2$ is a $p$-adic integer for any $n\in{\Bbb Z}^+$.

$\rm(ⅲ)$ Let $p>11$ be a prime. Then

where $x$ and $y$ are integers.

Remark 7.2. Note that the imaginary quadratic field ${\Bbb Q}(\sqrt{-330})$ has class number $8$.

Now we pose a conjecture related to the identity $(7.10)$.

Conjecture 7.4. $\rm(ⅰ)$ For any $n\in{\Bbb Z}^+$, we have

$\rm(ⅱ)$ Let $p>7$ be a prime with $p\not = 17$. Then

If $( \frac p7) = ( \frac p{17})$, then

is a $p$-adic integer for any $n\in{\Bbb Z}^+$.

$\rm(ⅲ)$ Let $p>7$ be a prime with $p\not = 17$. Then

where $x$ and $y$ are integers.

Remark 7.3. Note that the imaginary quadratic field ${\Bbb Q}(\sqrt{-357})$ has class number $8$.

Now we pose a conjecture related to the identity $(7.12)$.

Conjecture 7.5. $\rm(ⅰ)$ For any $n\in{\Bbb Z}^+$, we have

and this number is odd if and only if $n\in\{2^a:\ a\in{\Bbb N}\}$.

$\rm(ⅱ)$ Let $p>7$ be a prime. Then

If $( \frac {-14}p) = 1$, then

divided by $(pn)^2$ is a $p$-adic integer for each $n\in{\Bbb Z}^+$.

$\rm(ⅲ)$ Let $p>11$ be a prime. Then

where $x$ and $y$ are integers.

Remark 7.4. Note that the imaginary quadratic field ${\Bbb Q}(\sqrt{-462})$ has class number $8$. We believe that $462$ is the largest positive squarefree number $d$ for which the imaginary quadratic field ${\Bbb Q}(\sqrt{-d})$ can be used to construct a Ramanujan-type series for $1/\pi$.

The identities $(7.5),\,(7.7),\,(7.9),\,(7.11)$ are related to the imaginary quadratic fields ${\Bbb Q}(\sqrt{-165})$, ${\Bbb Q}(\sqrt{-210})$, ${\Bbb Q}(\sqrt{-345})$, ${\Bbb Q}(\sqrt{-330})$ (with class number $8$) respectively. We also have conjectures on related congruences similar to Conjectures 7.2, 7.3, 7.4 and 7.5.

To conclude this section, we confirm an open series for $1/\pi$ conjectured by the author (cf. [34,(3.28)] and [35,Conjecture 7.9]) in 2011.

Theorem 7.1. We have

where

Proof. The Franel numbers of order $4$ are given by $f_n^{(4)} = \sum_{k = 0}^n \binom nk^4\ (n\in{\Bbb N})$. Note that

By [11,(8.1)], for $|x|<1/16$ and $a,b\in{\Bbb Z}$, we have

Since

putting $a = 16$, $b = 5$ and $x = -1/20$ in (7.30) we obtain

As

by Cooper [9], we finally get

This concludes the proof of (7.29).

8.   Series and congruences involving $T_n(b,c)$ and $\beta_n=\sum_{k=0}^n \binom nk^2 \binom{n+k}k$

Recall that the numbers

are a kind of Apéry numbers. Let $p$ be an odd prime. For any $k = 0,1,\ldots,p-1$, we have

by [24,Lemma 2.7(ⅰ)]. Combining this with Remark 1.3(ⅱ), we see that

for any $b,c,m\in{\Bbb Z}$ with $p\nmid (b^2-4c)m$.

Wan and Zudilin [49] obtained the following irrational series for $1/\pi$ involving the Legendre polynomials and the numbers $\beta_n$:

Using our congruence approach (including Conjecture 1.4), we find one rational series for $1/\pi$ involving $T_n(b,c)$ and the Apéry numbers $\beta_n$ (see (8.1) below); Theorem 1 of [49] might be helpful to solve it.

Conjecture 8.1. (ⅰ) We have

Also, for any $n\in{\Bbb Z}^+$ we have

$\rm(ⅱ)$ Let $p>5$ be a prime. Then

If $p{\equiv}1\ ({\rm{mod}}\ 4)$, then

for all $n\in{\Bbb Z}^+$.

$\rm(ⅲ)$ Let $p>5$ be a prime. Then

Remark 8.1. This conjecture was formulated by the author on Oct. 27, 2019.

Conjecture 8.2. $\rm(ⅰ)$ For any $n\in{\Bbb Z}^+$, we have

and this number is odd if and only if $n\in\{2^a:\ a\in{\Bbb Z}^+\}$.

$\rm(ⅱ)$ Let $p$ be a prime. Then

If $( \frac {-15}p) = 1\ ($i.e., $p{\equiv}1,2,4,8\ ({\rm{mod}}\ {15}))$, then

divided by $(pn)^2$ is a $p$-adic integer for any $n\in{\Bbb Z}^+$.

$\rm(ⅲ)$ For any prime $p>5$, we have

Remark 8.2. This conjecture was formulated by the author on Nov. 13, 2019.

Conjecture 8.3. $\rm(ⅰ)$ For any $n\in{\Bbb Z}^+$, we have

and this number is odd if and only if $n$ is a power of two.

$\rm(ⅱ)$ Let $p>3$ be a prime. Then

If $p{\equiv}1\ ({\rm{mod}}\ 4)$, then

for all $n\in{\Bbb Z}^+$.

$\rm(ⅲ)$ For any odd prime $p$, we have

Remark 8.3. This conjecture was formulated by the author on Nov. 13, 2019.

Conjecture 8.4. $\rm(ⅰ)$ For any $n\in{\Bbb Z}^+$, we have

and this number is odd if and only if $n\in\{2^a:\ a = 0,2,3,4,\ldots\}$.

$\rm(ⅱ)$ Let $p$ be any odd prime. Then

If $p{\equiv}\pm1\ ({\rm{mod}}\ 8)$, then

for all $n\in{\Bbb Z}^+$.

$\rm(ⅲ)$ For any odd prime $p$, we have

Conjecture 8.5. $\rm(ⅰ)$ For any $n\in{\Bbb Z}^+$, we have

$\rm(ⅱ)$ Let $p>3$ be a prime. Then

If $p{\equiv}1\ ({\rm{mod}}\ 4)$, then

for all $n\in{\Bbb Z}^+$.

$\rm(ⅲ)$ Let $p\not = 2,5$ be a prime. Then

Conjecture 8.6. $\rm(ⅰ)$ For any $n\in{\Bbb Z}^+$, we have

and this number is odd if and only if $n$ is a power of two.

$\rm(ⅱ)$ Let $p$ be an odd prime. Then

If $( \frac{33}p) = 1$, then

for all $n\in{\Bbb Z}^+$.

$\rm(ⅲ)$ Let $p>3$ be a prime. Then

Conjecture 8.7. $\rm(ⅰ)$ For any $n\in{\Bbb Z}^+$, we have

and this number is odd if and only if $n$ is a power of two.

$\rm(ⅱ)$ Let $p>5$ be a prime. Then

If $( \frac{-15}p) = 1$, then

for all $n\in{\Bbb Z}^+$.

$\rm(ⅲ)$ For any prime $p>7$, we have

9.   Series and congruences involving $w_n=\sum_{k=0}^{\lfloor n/3\rfloor}(-1)^k3^{n-3k} \binom n{3k} \binom{3k}k \binom{2k}k$ and $T_n(b,c)$

The numbers

were first introduced by Zagier [51] during his study of Apéry-like integer sequences, who noted the recurrence

Lemma 9.1. Let $p>3$ be a prime. Then

Proof. Note that

with the help of the known congruence $\sum_{k = 0}^{p-1} \binom{2k}k \binom{3k}k/27^k{\equiv}( \frac p3)\ ({\rm{mod}}\ {p^2})$ conjectured by F. Rodriguez-Villegas [28] and proved by E. Mortenson [25]. Similarly,

By induction,

for all $n\in{\Bbb N}$. In particular,

So we have $w_k{\equiv}( \frac{-3}p)27^kw_{p-1-k}\ ({\rm{mod}}\ p)$ for $k = 0,1$. (Note that $w_0 = 1$ and $w_1 = 3$.)

Now let $k\in\{1,\ldots,p-2\}$ and assume that

Then

and hence

In view of the above, we have proved the desired result by induction.

For Lemma 9.1 one may also consult [31,Corollary 3.1]. Let $p>3$ be a prime. In view of Lemma 9.1 and Remark 1.3(ⅱ), we have

for any $b,c,m\in{\Bbb Z}$ with $p\nmid (b^2-4c)m$.

Wan and Zudilin [49] obtained the following irrational series for $1/\pi$ involving the Legendre polynomials and the numbers $w_n$:

Using our congruence approach (including Conjecture 1.4), we find five rational series for $1/\pi$ involving $T_n(b,c)$ and the numbers $w_n$; Theorem 1 of [49] might be helpful to solve them.

Conjecture 9.1. We have

Below we present our conjectures on congruences related to the identities (9.2) and (9.5).

Conjecture 9.2. $\rm(ⅰ)$ For any $n\in{\Bbb Z}^+$, we have

and this number is odd if and only if $n\in\{2^a:\ a\in{\Bbb N}\}$.

$\rm(ⅱ)$ Let $p>3$ be a prime. Then

If $( \frac p7) = 1\ ($i.e., $p{\equiv}1,2,4\ ({\rm{mod}}\ 7))$, then

for all $n\in{\Bbb Z}^+$.

$\rm(ⅲ)$ For any prime $p>7$, we have

Conjecture 9.3. $\rm(ⅰ)$ For any $n\in{\Bbb Z}^+$, we have

and this number is odd if and only if $n\in\{2^a:\ a\in{\Bbb N}\}$.

$\rm(ⅱ)$ Let $p>3$ be a prime with $p\not = 7$. Then

If $( \frac p7) = 1\ ($i.e., $p{\equiv}1,2,4\ ({\rm{mod}}\ 7))$, then

for all $n\in{\Bbb Z}^+$.

$\rm(ⅲ)$ For any prime $p>3$ with $p\not = 7,13$, we have

Now we give one more conjecture in this section.

Conjecture 9.4. $\rm(ⅰ)$ For any integer $n>1$, we have

$\rm(ⅱ)$ Let $p>3$ be a prime. Then

If $p{\equiv}1\ ({\rm{mod}}\ 4)$, then

for all $n\in{\Bbb Z}^+$.

$\rm(ⅲ)$ For any prime $p>3$, we have

Remark 9.1. For primes $p>3$ with $p{\equiv}3\ ({\rm{mod}}\ 4)$, in general the congruence (9.16) is not always valid for all $n\in{\Bbb Z}^+$. This does not violate Conjecture 1.2 since $\lim_{k\to+\infty}|w_kT_k(10,-2)|^{1/k} = \sqrt{27}\times\sqrt{10^2-4(-2)} = 54$. If the series $\sum_{k = 0}^\infty \frac{2k+1}{54^k}w_kT_k(10,-2)$ converges, its value times $\pi/\sqrt3$ should be a rational number.

10.   Series for $\pi$ involving $T_n$ and related congruences

Let $p$ be an odd prime and let $a,b,c,d,m\in{\Bbb Z}$ with $m(b^2-4c)\not{\equiv}0\ ({\rm{mod}}\ p)$. Then

with the aid of [33,Lemma 2.1]. Thus

in view of Remark 1.3(ⅱ).

Let $p>3$ be a prime. By the above, the author's conjectural congruence (cf. [35,Conjecture 1.3])

implies that

Motivated by this, we pose the following curious conjecture.

Conjecture 10.1. We have the following identities:

Remark 10.1. The two identities were conjectured by the author on Dec. 7, 2019. One can easily check them numerically via $\mathsf{Mathematica}$ as the two series converge fast.

Now we state our related conjectures on congruences.

Conjecture 10.2. For any prime $p>3$, we have

and

Conjecture 10.3. (ⅰ) We have

for all $n\in{\Bbb Z}^+$, and also

for each prime $p>3$.

$\rm(ⅱ)$ For any prime $p{\equiv}1\ ({\rm{mod}}\ 3)$ and $n\in{\Bbb Z}^+$, we have

Remark 10.2. See also [45,Conjecture 67] for a similar conjecture.

Let $p$ be an odd prime. We conjecture that

and

Though (10.6) implies the congruence

and (10.7) with $p>3$ implies the congruence

we are unable to find the exact values of the two converging series

Acknowledgment

The author would like to thank Prof. Qing-Hu Hou at Tianjin Univ. for his helpful comments on the proof of Lemma 2.3.