Russian President Vladimir Putin is hosting Turkish counterpart Recep Tayyip Erdogan in their first meeting since Turkey shot down a Russian fighter jet last November, sparking a deep diplomatic crisis.

Erdogan's visit to Putin's hometown of St Petersburg on Tuesday is also his first foreign trip since a failed coup attempt last month that sparked a purge of alleged coup supporters in the military, judiciary, civil service and education sector, and cast a shadow over Turkey's relations with the West.

"Your visit today, despite a very difficult situation regarding domestic politics, indicates that we all want to restart dialogue and restore relations between Russia and Turkey," Putin said as the two met.

Erdogan said that Turkey was entering a "very different period" in relations with Russia, and that solidarity between the two countries would help the resolution of regional problems.

Erdogan also thanked his Russian counterpart for a telephone call after the failed coup attempt on July 15, saying it "brought our people great happiness".

The shooting down of the Russian jet by a Turkish F-16 over the Syrian border last November saw a furious Putin slap economic sanctions on Turkey and launch a blistering war of words with Erdogan that seemed to irrevocably damage burgeoning ties.

"When the Turkish military shot down a Russian fighter jet that it said strayed from Syrian into Turkish airspace last November, Moscow's retaliation was swift," Al Jazeera's Bernard Smith, reporting from Istanbul, said.

OPINION: Why Turkey is mending ties with old foes

"Tourist charter flights to Turkey stopped, Russian visitor numbers fell by 87 percent. Turkey's exports to Russia, including food, fell by more than half to $730m in the first six months of this year."

But, in a reversal in late June, Putin accepted a personal expression of regret over the incident from Erdogan as an apology, immediately rolled back a ban on the sale of package holidays to Turkey and signalled Moscow would end measures against food imports and construction firms from the country.

Now, following the failed coup attempt, analysts say ties between the two could deepen - with Erdogan publicly making it clear he feels let down by the United States and the European Union.

READ MORE - Erdogan to West: 'Mind your own business'

"The Russian President was much quicker in his condemnation of the attempted coup than many of Turkey's Western allies and they've [the Western allies] also expressed alarm at the extent of the post-coup crackdown," said Al Jazeera's Smith. "Vladimir Putin hasn't got involved."

Syria is expected to be high on the agenda during the visit. Moscow's military support for Syria's President Bashar al-Assad has been credited for helping to keep him in power.

Turkey, though, wants him gone.

"Currently Turkey cannot enter Syria, it cannot do anything in Syria because the Russian forces are there," Erhan Ersan, a Russian affairs specialist at Istanbul's Marmara University, told Al Jazeera.

"In order to solve that, they need to bring their positions closer to Russia. If you hear the messages coming from President Erdogan he's actually open to that. He wants to build a new framework of relations based on Syria with Russia as well."

READ MORE: 'Which parts of Syria should be bombed?'

The head of the opposition Syrian National Coalition welcomed the meeting.

Speaking at a press conference in Istanbul on Monday, Anas al-Abda said Erdogan’s visit could be a "positive step" for finding a solution to a ruinous war that has killed hundreds of thousands.

“We consider the Turkish president as a key ally of the Syrian people," al-Abda told Turkey's state-run Anadolu Agency. "He has a chance to propose ideas and initiatives to Russians and to explain them the current situation in Syria."